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GNDU QUESTION PAPERS 2021
BA/BSc 4
th
SEMESTER
QUANTITATIVE TECHNIQUES – IV
Time Allowed: 3 Hours Maximum Marks:
Note: Aempt Five quesons in all, selecng at least One queson from each secon. The
Fih queson may be aempted from any secon. All quesons carry equal marks.
1. The following data relates to three variables
and
; obtain the equaon of the
plane of regression of
on
and
. Also esmate value of
when
and
.
X₁
4
6
7
9
13
15
X₂
15
12
8
6
4
3
X₃
30
24
20
14
10
4
2.(a) Dierenate between paral and mulple correlaon coecients.
(b) Discuss the procedure to esmate modied exponenal curve.
(c) Fit exponenal curve of type
to the following data :
x
1
2
3
4
5
y
1.6
4.5
13.8
40.2
125.0
3.(a) Dene probability. Also explain laws of addion and mulplicaon.
(b) A problem in stascs is given to three students A, B and C whose chances of solving it
are 1/3, 1/4 and 1/5 respecvely. Find the probability that the problem will be solved if
they all try independently.
4.(a) Dene random variable. What is probability density funcon ?
(b) An unbiased coin is tossed 3 mes. If a random variable ‘X’ is dened as number of
heads; then nd probability mass funcon of X.
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(c) What do you mean by mathemacal expectaon ? Also discuss some important
properes of mathemacal expectaon.
5. Dene a binomial variate with parameters n and p and obtain its probability funcon.
Also derive important properes of binomial distribuon.
6. What is normal distribuon ? Draw a rough sketch of its probability density funcon.
Also derive its moment generang funcon.
7.(a) Disnguish between populaon and sample. Also discuss important features of a
good sample.
(b) Write a note on the concept of standard error of esmates.
8. Disnguish between random and subjecve sampling. What is simple random sampling
? Discuss its merits and limitaons.
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GNDU ANSWER PAPERS 2021
BA/BSc 4
th
SEMESTER
QUANTITATIVE TECHNIQUES – IV
Time Allowed: 3 Hours Maximum Marks:
Note: Aempt Five quesons in all, selecng at least One queson from each secon. The
Fih queson may be aempted from any secon. All quesons carry equal marks.
1. The following data relates to three variables
and
; obtain the equaon of the
plane of regression of
on
and
. Also esmate value of
when
and
.
X₁
4
6
7
9
13
15
X₂
15
12
8
6
4
3
X₃
30
24
20
14
10
4
Ans: Regression plane of X₁ on X₂ and X₃
Imagine you’re trying to predict a student’s performance (X₁) based on two study habits:
hours of practice (X₂) and the number of revision sessions (X₃). You suspect X₁ depends on
both X₂ and X₃, but not in a perfectly simple way. Multiple regression is your way of drawing
a “best-fit plane” through the cloud of points in three-dimensional space, so you can
estimate X₁ for any combination of X₂ and X₃.
Below is the actual dataset:
• X₁: 4, 6, 7, 9, 13, 15
• X₂: 15, 12, 8, 6, 4, 3
• X₃: 30, 24, 20, 14, 10, 4
What we mean by the regression plane
When you regress X₁ on X₂ and X₃, you’re finding the plane:
• a is the intercept (where the plane crosses the X₁-axis when X₂ = X₃ = 0).
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• b₂ tells you how much X₁ changes, on average, for each unit increase in X₂, holding X₃
constant.
• b₃ tells you the same for X₃, holding X₂ constant.
This plane minimizes the sum of squared vertical distances (errors) between the actual X₁
values and the values predicted by the plane.
Step-by-step narrative of how the plane is found
You can think of the process in two parallel ways—intuitively and procedurally.
Intuitive picture
• Plot the six points in 3D: each is at coordinates (X₂, X₃, X₁).
• Imagine adjusting a flat sheet (the plane) so that it sits “closest” to all points.
• The best-fit sheet is the regression plane: it balances the ups and downs of residuals.
Procedural steps
1. Set up the model:
o You assume a linear relationship:
where is the error term.
2. Use least squares to estimate a, b₂, b₃:
o In matrix terms, you solve the normal equations:
where is the design matrix with a column of ones (for the intercept), a column of X₂, and a
column of X₃; is the vector of X₁.
3. Compute the coefficients:
o From the data, the estimated regression plane comes out to:
• Interpretation:
o Intercept (16.4776): baseline level of X₁ when X₂ = X₃ = 0 (mostly interpretive,
since 0 may be outside your observed range).
o b₂ = 0.3899: increasing X₂ by 1 unit increases X₁ by about 0.39, if X₃ stays the
same.
o b₃ = −0.6233: increasing X₃ by 1 unit decreases X₁ by about 0.62, if X₂ stays
the same.
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It’s common to see one predictor positively related and another negatively related—this
simply means, within this dataset, holding the other constant, X₂ is associated with higher
X₁, while X₃ is associated with lower X₁.
Estimating X₁ when X₂ = 15 and X₃ = 30
Now plug the values into the plane:
(approximately)
Rounded to four decimals, the estimate is:
This is consistent with the pattern in the data: when X₂ is high and X₃ is very high, the plane
predicts a relatively low X₁ for this dataset’s structure.
Making sense of the signs and magnitudes
• Positive b₂ (0.3899): As X₂ increases (e.g., more of the factor represented by X₂), X₁
tends to increase, if X₃ doesn’t change.
• Negative b₃ (−0.6233): As X₃ increases, X₁ tends to decrease, if X₂ doesn’t change.
This can happen when two input variables pull the outcome in opposite directions or
when they’re inversely related to X₁ once you control for each other.
The magnitude tells you the strength of change per unit. Here, X₃’s coefficient is larger in
size than X₂’s, which means a unit change in X₃ has a bigger effect on X₁ (in the opposite
direction) than a unit change in X₂ (positive direction), all else equal.
A relatable analogy
Think of X₁ as “final score,” X₂ as “time spent solving practice problems,” and X₃ as “time
spent on distractions.” If you increase practice time (X₂), the score goes up a bit; if
distractions (X₃) increase, the score goes down even more. The plane is the formula that
blends both effects to predict the final score.
Quick checkpoints students often forget
• Centering matters for interpretation: If you center X₂ and X₃ (subtract their means),
the intercept becomes the predicted X₁ at average X₂ and X₃, which is often easier to
interpret.
• Extrapolation caveat: Predicting at values far outside the observed range can be
risky. Fortunately, here X₂ = 15 and X₃ = 30 are in your observed data (first row),
making the estimate sensible.
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• Correlation among predictors: If X₂ and X₃ are strongly related to each other
(multicollinearity), coefficients can flip signs or change magnitude. Always look at the
data structure.
Final answers
• Regression plane of X₁ on X₂ and X₃:
• Estimated value of X₁ when
and
:
2.(a) Dierenate between paral and mulple correlaon coecients.
(b) Discuss the procedure to esmate modied exponenal curve.
(c) Fit exponenal curve of type
to the following data :
x
1
2
3
4
5
y
1.6
4.5
13.8
40.2
125.0
Ans: (a) Difference between Partial and Multiple Correlation Coefficients
Imagine three friends: X, Y, and Z.
We want to understand how relationships work among them.
Multiple Correlation
Multiple correlation answers this question:
“How strongly is one variable related to two or more other variables together?”
For example:
• How much is Y affected by X and Z together?
It doesn’t try to isolate anything.
It simply asks: “If we use both X and Z, how well can we explain Y?”
So, multiple correlation is like saying:
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“If I consider both maths marks and study hours together, how well can I predict exam
performance?”
It treats the other variables as a team.
We usually write it like:
•
meaning correlation of Y with X and Z together
This value is always positive or zero, never negative.
Partial Correlation
Partial correlation is a bit more clever.
It answers this kind of question:
“What is the relationship between two variables after removing the effect of the third
one?”
Imagine you want to know:
• Is there a real relationship between X and Y
• Or is it just happening because of Z?
So partial correlation removes (or “controls”) the effect of another variable.
Example:
Suppose height, age, and weight are related.
If we want to know:
• How much are height and weight related independent of age?
We use partial correlation.
We usually write:
•
Meaning: correlation between X and Y, controlling Z.
Partial correlation can be positive or negative, because once the third effect is removed,
reality may change.
Simple human difference
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Partial Correlation
Multiple Correlation
Studies relationship between two
variables, removing the effect of
others
Studies relationship between one dependent
variable and multiple independent variables
together
Removes influence of other variables
Combines influence of other variables
Can be +ve or −ve
Always +ve or zero
Usually used to isolate true
relationships
Used to predict outcome using many factors
So think of it like:
• Multiple correlation = teamwork effect
• Partial correlation = real friendship after removing third-wheel influence
(b) Procedure to Estimate Modified Exponential Curve
The word "curve" may sound scary, but actually this is just about fitting a smooth
mathematical line to data that grows rapidly — like population growth, bacteria spreading,
or investment increasing with compound interest.
A Modified Exponential Curve is basically of the form:
Where:
• = starting value or base level
• = growth factor
• = time or independent variable
This curve is “exponential” because it doesn’t grow in a straight line.
It grows like:
• 2, 4, 8, 16…
• Or 5, 10, 20, 40…
So growth becomes faster and faster.
But how do we estimate it from real data?
Procedure (Step-by-Step Human Explanation)
Step 1: Write the given model
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Start with:
Step 2: Take logarithm
Why?
Because exponential curves are hard to work with in raw form, but taking log turns them
into a straight line.
Apply logarithm:
Let:
Now equation becomes:
Does this look familiar?
Yes! It is now a straight-line equation like
So now we can use ordinary least squares to estimate A and B.
Step 3: Use normal equations
We use:
Solve these like simple linear regression.
Step 4: Convert back
Once A and B are found:
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\antilog \antilog
Finally we get our required curve:
Boom! Exponential curve fitted
(c) Fit Exponential Curve to Given Data
We are given:
x
1
2
3
4
5
y
1.6
4.5
13.8
40.2
125
We need to fit:
Step 1: Take logs
Take natural log (or log base 10; any is fine as long as consistent):
x
y
ln(y)
1
1.6
0.470
2
4.5
1.504
3
13.8
2.625
4
40.2
3.695
5
125
4.828
Let:
Step 2: Treat it like linear regression
We already calculated:
• Mean of x = 3
• Mean of ln(y) ≈ 2.624
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Slope:
After calculation:
So:
Step 3: Find intercept
So:
Final Fitted Exponential Curve
This is the best-fitting exponential curve according to least squares.
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3.(a) Dene probability. Also explain laws of addion and mulplicaon.
(b) A problem in stascs is given to three students A, B and C whose chances of solving it
are 1/3, 1/4 and 1/5 respecvely. Find the probability that the problem will be solved if
they all try independently.
Ans: Probability, Laws of Addition and Multiplication, and a Practical Problem
Probability is one of those fascinating concepts that blends mathematics with everyday life.
Whether you’re tossing a coin, predicting the weather, or estimating exam success,
probability gives us a way to measure uncertainty. Let’s break this question down step by
step, so it feels simple, relatable, and enjoyable.
(a) Definition of Probability
Probability is the measure of how likely an event is to occur. It is expressed as a number
between 0 and 1 (or sometimes as a percentage).
• 0 probability means the event is impossible.
• 1 probability means the event is certain.
• Values in between represent varying degrees of likelihood.
Formula:
Number of favorable outcomes
Total number of possible outcomes
Example: If you toss a fair coin, the probability of getting heads is:
Heads
Laws of Probability
Probability has two fundamental laws that help us deal with multiple events: the law of
addition and the law of multiplication.
1. Law of Addition
This law applies when we want the probability of either one event or another event
happening.
• If two events are mutually exclusive (cannot happen at the same time), then:
or
• If events are not mutually exclusive, then:
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or and
Example:
• Probability of drawing a red card OR a king from a deck.
• Since some kings are red, we subtract the overlap.
2. Law of Multiplication
This law applies when we want the probability of two events happening together.
• If events are independent (one does not affect the other):
and
• If events are dependent, then:
and
Example:
• Probability of tossing a coin and getting heads and rolling a die and getting a 6.
Heads and 6
(b) Problem with Three Students
Now let’s solve the actual problem:
Three students A, B, and C attempt a statistics problem independently. Their chances of
solving it are:
• A:
• B:
• C:
We want the probability that the problem will be solved if they all try independently.
Step 1: Probability of NOT solving the problem
It’s often easier to calculate the probability that none of them solves the problem, and then
subtract from 1.
• Probability A fails =
• Probability B fails =
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• Probability C fails =
Step 2: Probability all three fail
Since they work independently:
All fail
All fail
Step 3: Probability at least one solves
Now subtract from 1:
At least one solves All fail
At least one solves
So, the probability that the problem will be solved is 3/5 or 0.6 (60%).
A Relatable Story
Imagine three friends—Aman, Bhavna, and Chandan—are given a tricky statistics problem.
Aman is fairly good, with a 1/3 chance of solving it. Bhavna is a bit weaker, with a 1/4
chance. Chandan struggles the most, with only a 1/5 chance.
If they all try independently, what’s the chance that at least one of them succeeds?
• First, think of the worst case: all three fail. That’s like multiplying their failure
chances:
.
• So, there’s a 40% chance they all fail.
• Which means there’s a 60% chance that at least one succeeds.
This is encouraging! Even though each student individually has a modest chance, together
they have a strong chance of success.
Why This Matters
This problem shows how probability helps us understand teamwork and independence:
• Individually, each student has limited chances.
• Collectively, their chances improve because probability accounts for “at least one
success.”
• This is why group problem-solving often works better than working alone.
Summary Table
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Student
Chance of Success
Chance of Failure
A
1/3
2/3
B
1/4
3/4
C
1/5
4/5
All fail
At least one solves
Final Thoughts
Probability is not just abstract math—it’s a way of thinking about uncertainty in real life.
• The definition of probability gives us a measure of likelihood.
• The laws of addition and multiplication help us combine events logically.
• The problem with three students shows how teamwork improves chances: even if
each has a small probability, together they have a 60% chance of success.
So, probability teaches us optimism: when independent efforts combine, the chances of
success rise. Next time you face a tough problem, remember—you don’t have to solve it
alone. Mathematics itself proves that collaboration increases the odds of victory.
4.(a) Dene random variable. What is probability density funcon ?
(b) An unbiased coin is tossed 3 mes. If a random variable ‘X’ is dened as number of
heads; then nd probability mass funcon of X.
(c) What do you mean by mathemacal expectaon ? Also discuss some important
properes of mathemacal expectaon.
Ans: (a) Random Variable and Probability Density Function – Simple Meaning
First, think about real-life situations involving uncertainty. For example, you throw a dice,
toss a coin, measure someone’s height, or calculate how long a bus takes to arrive. These
situations do not have a fixed outcome; they have different possible outcomes. To study
these outcomes mathematically, we give them numbers.
That number is called a Random Variable.
A Random Variable is simply a rule that assigns a number to every possible outcome of a
random experiment.
For example:
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• Toss a coin → you may get Head or Tail
If we say:
Head = 1
Tail = 0
Then “number showing Head or Tail” becomes a random variable.
Another example:
Toss a coin 3 times and count how many heads appear. The number of heads (0, 1, 2, or 3) is
also a random variable.
So don’t be scared by the word “variable” or “random.” It just means:
A value that changes depending on the outcome of a random process.
Now let's move to Probability Density Function (PDF).
Random variables are of two types:
1. Discrete Random Variable – Takes specific separate values (like 0,1,2,3 etc.)
2. Continuous Random Variable – Can take infinitely many values in an interval (like
height, time, weight, etc.)
For continuous random variables, we use Probability Density Function (PDF).
A Probability Density Function tells us how probability is distributed over different values of
a continuous random variable.
You cannot directly say “the probability that height is exactly 5.5 ft,” because there are
infinite possible heights. So instead, we talk about the probability in an interval, like:
• Probability that height lies between 5.4 and 5.6
• Probability that waiting time is between 5 and 10 minutes
PDF must satisfy two conditions:
It is never negative
The total area under the curve is equal to 1 (meaning total probability is 1)
So, in simple words:
A random variable gives numerical meaning to random events.
A probability density function describes how probability is spread over a range of
continuous values.
(b) Probability Mass Function for Coin Toss Example
Now let’s solve the given part (b) in the easiest way possible.
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We toss an unbiased coin 3 times.
A random variable X = Number of Heads obtained.
First, list all possible outcomes.
When you toss 3 times, total number of outcomes:
2 × 2 × 2 = 8
They are:
HHH
HHT
HTH
THH
HTT
THT
TTH
TTT
Now count number of heads in each:
• 3 Heads → HHH → X = 3
• 2 Heads → HHT, HTH, THH → X = 2 (3 outcomes)
• 1 Head → HTT, THT, TTH → X = 1 (3 outcomes)
• 0 Heads → TTT → X = 0
So X can take values:
0, 1, 2, 3
Now calculate probability of each value.
Since the coin is unbiased, every outcome has probability:
1/8
So,
• P(X = 0) = 1/8
• P(X = 1) = 3/8
• P(X = 2) = 3/8
• P(X = 3) = 1/8
This table is called Probability Mass Function (PMF).
PMF simply tells:
What values a discrete random variable can take
And what probability belongs to each value
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Also check:
Total probability = 1/8 + 3/8 + 3/8 + 1/8 = 8/8 = 1
So it is correct.
(c) Mathematical Expectation – Meaning and Properties
Now let us understand Mathematical Expectation, also called Expectation, Mean, or
Expected Value.
Imagine you are playing a game where you toss a coin and get ₹1 for every head. If you play
many many times, what average amount will you win per game? That long-run average
value is called Expectation.
So, in simple words:
Mathematical Expectation is the average value of a random variable in the long run.
For a discrete random variable, it is calculated as:
E(X) = Σ [x × P(x)]
That means:
Multiply each value of X by its probability, then add all.
For example, from our coin case:
X: 0, 1, 2, 3
P(X): 1/8, 3/8, 3/8, 1/8
So,
E(X) = 0×1/8 + 1×3/8 + 2×3/8 + 3×1/8
= 0 + 3/8 + 6/8 + 3/8
= 12/8
= 1.5
So the expected number of heads when we toss a coin 3 times is 1.5.
Of course, you will never actually get “1.5 heads,” but expectation represents the average
behavior in the long run.
Important Properties of Mathematical Expectation
Let us discuss them in a simple and human way.
1. Linearity Property
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If you multiply a random variable by a constant or add something to it, expectation behaves
just like normal algebra.
If Y = aX + b
Then,
E(Y) = aE(X) + b
Meaning:
Expectation of scaled or shifted variable changes in the same style.
2. Expectation of Sum
If you have two random variables X and Y:
E(X + Y) = E(X) + E(Y)
This is beautiful because it does not require independence. It always holds true.
3. Expectation of a Constant
If C is a constant:
E(C) = C
A fixed value is always itself on average.
4. Non-Negativity (for non-negative variables)
If X never takes negative values, its expectation cannot be negative.
5. Expectation Represents Long-Run Average
If you repeat an experiment many times, the average result will approach expectation. This
connects probability with real life.
Conclusion
So, the whole question connects beautifully:
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• A Random Variable converts uncertain events into numbers.
• A Probability Density Function shows how probability is spread for continuous
values.
• The Probability Mass Function gives probabilities of different values for discrete
variables, like number of heads in our coin example.
• Mathematical Expectation tells us the long-term average outcome and follows some
very useful mathematical properties.
5. Dene a binomial variate with parameters n and p and obtain its probability funcon.
Also derive important properes of binomial distribuon.
Ans: What Is a Binomial Variate?
A binomial variate is a random variable that counts the number of successes in a fixed
number of independent trials, where each trial has only two outcomes.
Let’s simplify that!
There are four basic conditions for something to be called a binomial situation:
Fixed Number of Trials (n)
You must decide in advance how many times the experiment will be repeated.
Example: tossing a coin 10 times → n = 10.
Each Trial Has Only Two Outcomes
They are usually called:
• Success
• Failure
For example:
• Head (success) / Tail (failure)
• Pass / Fail
• Hit / Miss
Constant Probability of Success (p)
The chance of success remains the same in every trial.
Example:
Probability of getting heads on every coin toss is always 1/2.
Then probability of failure is:
p = probability of success
q = probability of failure
So,
q = 1 − p
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Trials Are Independent
Result of one trial does not affect another.
Example:
• Tossing a coin multiple times
• Checking each bulb independently
So, the binomial variate X = number of successes in n trials.
We write:
X ~ Binomial (n, p)
This simply means:
“X follows a binomial distribution with n trials and probability p of success.”
Probability Function of Binomial Distribution
Now comes the most interesting question:
“What is the probability of getting exactly x successes in n trials?”
Imagine tossing a coin 5 times. What is the chance of getting exactly 2 heads?
To answer such questions, we use the Binomial Probability Function.
Step-by-Step Idea
Suppose:
• n = number of trials
• p = probability of success
• q = 1 − p = probability of failure
• X = number of successes we want
• x = specific number of successes
To get x successes, we must also get (n − x) failures.
Probability of one specific arrangement of x successes and (n − x) failures is:
pˣ × qⁿ⁻ˣ
But wait! Successes can occur in many different arrangements.
For example:
If we toss a coin three times and want exactly 2 heads, possible outcomes are:
• H H T
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• H T H
• T H H
There are multiple ways.
So, we must multiply by the number of ways of choosing x successes out of n trials.
That is:
So the Binomial Probability Function is:
Where:
• x = 0,1,2,3,...,n
This tells us the probability of exactly x successes in n trials.
This is the heart of binomial distribution.
Important Properties of the Binomial Distribution
Now let’s understand the key features that make binomial distribution meaningful and
useful.
Mean (Expected Value)
The mean tells us:
“What result do we expect on average?”
For binomial distribution:
Mean
Example:
If a batsman has 40% chance of hitting a boundary per ball (p = 0.4),
and he faces 6 balls:
Mean = np = 6 × 0.4 = 2.4
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So on average, he is expected to hit around 2 to 3 boundaries.
Variance and Standard Deviation
Variance measures spread — how much the values vary around the mean.
For binomial distribution:
Variance
Standard deviation is simply the square root:
If p is near 0.5, variance is larger because results are more spread out.
If p is near 0 or 1, variance is smaller because outcomes are more predictable.
Shape of Binomial Distribution
The shape depends on p:
✔ If p = 0.5
Distribution is symmetric — equal chance of success and failure.
✔ If p > 0.5
Distribution is positively skewed (more successes likely)
✔ If p < 0.5
Distribution is negatively skewed (more failures likely)
As n becomes large, binomial distribution starts looking like the normal distribution. This is
a powerful concept used in statistics.
Range of X
X can take values:
0, 1, 2, 3, ..., n
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Meaning:
Minimum successes = 0
Maximum successes = n
Moment Generating Function (Just for Understanding)
Though not required in deep detail at student level, still:
Moment generating function helps derive moments like mean and variance.
For binomial:
From this, statisticians derive many properties.
Why Binomial Distribution Is So Useful?
Because real life is full of yes/no situations, like:
• Will it rain today? Yes / No
• Will a student pass? Yes / No
• Is a product defective? Yes / No
• Will a customer buy a product? Yes / No
Whenever such events repeat multiple times, binomial distribution becomes the perfect
mathematical tool.
6. What is normal distribuon ? Draw a rough sketch of its probability density funcon.
Also derive its moment generang funcon.
Ans: Normal distribution: definition, PDF sketch, and MGF derivation
Imagine measuring the heights of a large group of students from Amritsar. Most will cluster
around the average height, fewer will be very short or very tall, and the distribution will look
beautifully symmetric. That bell-shaped pattern is the hallmark of the normal distribution —
the most widely used model in statistics when data arise from many small, additive
influences.
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Definition of the normal distribution
The normal (Gaussian) distribution is a continuous probability distribution characterized by
two parameters: mean (the center) and standard deviation (the spread). A random
variable is normally distributed if its probability density function (PDF) is
• The curve is symmetric about .
• The spread of the curve is controlled by ; larger means a wider, flatter bell.
• The total area under the curve is 1, representing total probability.
Two special cases are common:
• The standard normal: , .
• A general normal: any real and positive .
Probability density function and a rough sketch
For the standard normal distribution, the PDF is
Here’s a rough sketch of the bell curve (standard normal), centered at 0 and tapering
towards both tails:
Key features to notice:
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• Peak at (here, 0).
• Symmetry: .
• Tails approach zero smoothly but never actually reach it.
A general normal with mean simply shifts this shape horizontally, and changing stretches
or squeezes the curve.
Why the normal distribution shows up so often
• Many measurements are influenced by numerous small, independent “nudges” (e.g.,
biological, environmental, or sampling variations). By the central limit theorem, the
sum of many small independent effects tends to look normal.
• Normality provides tractable mathematics (closed-form expressions for many
quantities), making it a convenient and powerful approximation in practice.
Deriving the moment generating function (MGF)
The moment generating function captures all moments of a distribution and is defined as
We’ll derive
for the normal
.
Step 1: Write the integral
Combine exponents:
Step 2: Complete the square in the exponent
Focus on the term inside the exponent:
Rewrite as :
So the exponent becomes:
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Now complete the square in
:
Therefore, the whole exponent simplifies to:
Step 3: Factor out the constants and change variables
So the integrand is
Thus,
Let
and the integral becomes the familiar standard normal integral:
Step 4: Final result
Hence,
This is the MGF of
. As a check, you can recover moments:
• Mean:
.
• Variance:
.
Interpreting the MGF
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• The exponential form
neatly encodes the first and second
moments; higher derivatives yield higher moments.
• It also reveals that the sum of independent normals is normal: if
and
, independent, then the MGF of
multiplies to another
Gaussian form, giving
.
A quick, concrete example
Suppose
represents adult heights (in cm). The PDF bell curve centers at 170
with spread controlled by . The MGF is
which confirms mean and variance . If you average many independent
measurements, the result trends toward normal — that’s why the bell curve is so common
in physical, biological, and social sciences.
7.(a) Disnguish between populaon and sample. Also discuss important features of a
good sample.
(b) Write a note on the concept of standard error of esmates.
Ans: Population, Sample, and Standard Error of Estimates
Statistics often feels like a bridge between raw data and meaningful conclusions. But to
cross that bridge safely, we need to understand two fundamental ideas: population and
sample. Once we grasp these, we can talk about what makes a sample “good” and why the
concept of standard error is so important in estimation.
(a) Population vs. Sample
1. Population
In statistics, a population refers to the entire set of individuals, items, or data points that we
are interested in studying.
• It is the “whole universe” of data.
• Populations can be finite (like all students in a school) or infinite (like all possible
outcomes of rolling a die).
• Parameters such as population mean (μ) and population variance (σ²) describe the
population.
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Example: If you want to study the average height of students in Amritsar University, the
population is all students enrolled in the university.
2. Sample
A sample is a subset of the population, selected for actual study.
• Since studying the entire population is often impractical, we collect data from a
smaller group.
• Statistics like sample mean () and sample variance (s²) describe the sample.
• The goal is to use the sample to make inferences about the population.
Example: Instead of measuring every student’s height, you might measure 100 randomly
chosen students. That group of 100 is your sample.
Key Distinction
• Population: The complete set (the “whole cake”).
• Sample: A part of the set (a “slice of the cake”).
• Population gives parameters; sample gives statistics.
• Sampling is necessary because populations are often too large to study directly.
Features of a Good Sample
Not all samples are equally useful. A good sample must represent the population fairly. Here
are the important features:
1. Representativeness
o The sample should reflect the characteristics of the population.
o If the population has equal numbers of men and women, the sample should
too.
2. Adequate Size
o The sample must be large enough to capture diversity.
o Too small a sample may give misleading results.
3. Random Selection
o Every member of the population should have an equal chance of being
selected.
o Random sampling reduces bias.
4. Freedom from Bias
o A sample should not favor certain groups.
o For example, surveying only morning students would bias results if evening
students differ.
5. Practicality
o The sample should be feasible to collect in terms of time, cost, and effort.
6. Accuracy and Reliability
o A good sample produces results close to the true population values.
o Repeated sampling should give consistent outcomes.
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Example: If you want to know the average income of families in Punjab, a good sample
would include families from rural and urban areas, different occupations, and income levels.
A biased sample (say, only urban families) would misrepresent the population.
(b) Standard Error of Estimates
Now let’s move to the second part: standard error.
1. What is Standard Error?
The standard error (SE) measures the variability of a sample statistic (like the sample mean)
from sample to sample.
• It tells us how much the sample estimate is likely to differ from the true population
parameter.
• In simple words: SE is the “margin of uncertainty” in our estimate.
Formula for the standard error of the mean:
where:
• = population standard deviation
• = sample size
If is unknown, we use the sample standard deviation (s).
2. Why Standard Error Matters
• Precision of Estimates: Smaller SE means the sample mean is closer to the
population mean.
• Confidence Intervals: SE helps construct confidence intervals around estimates.
• Hypothesis Testing: SE is used to calculate test statistics (like t-tests and z-tests).
Example: Suppose the average height of a sample of 100 students is 165 cm, with SE = 2
cm. This means the true population mean is likely to be close to 165 cm, within a margin of
about 2 cm.
3. Factors Affecting Standard Error
• Sample Size (n): Larger samples reduce SE because variability averages out.
• Population Variability (σ): Greater variability increases SE.
• Sampling Method: Random, unbiased samples reduce SE compared to biased ones.
4. A Relatable Story
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Imagine you’re baking cookies and want to know the average weight of cookies in the batch.
Instead of weighing all 500 cookies, you weigh 20 randomly chosen ones.
• The average weight of your sample is 25 grams.
• But you know not all cookies are identical—some are slightly heavier or lighter.
• The standard error tells you how much your sample’s average (25 grams) might
differ from the true average of all 500 cookies.
If SE is small (say, 0.5 grams), you can be confident your estimate is close. If SE is large (say,
3 grams), your estimate is less reliable.
Putting It All Together
• Population vs. Sample: Population is the whole group; sample is a part of it.
• Good Sample Features: Representativeness, adequate size, randomness, freedom
from bias, practicality, and reliability.
• Standard Error: A measure of how much sample estimates vary from the true
population parameter.
Together, these concepts form the backbone of statistical inference. Without good samples,
our estimates are unreliable. Without understanding standard error, we cannot judge how
trustworthy our estimates are.
Summary Table
Concept
Meaning
Example
Population
Entire group under study
All students in a university
Sample
Subset of population used for study
100 randomly chosen students
Good Sample
Representative, unbiased, adequate
size
Survey including rural + urban
families
Standard
Error
Variability of sample estimate
SE of mean height = 2 cm
Final Thoughts
Statistics is about making sense of the world with limited information. We rarely have
access to the entire population, so we rely on samples. But not just any sample will do—it
must be representative, unbiased, and reliable. Once we have a good sample, the standard
error tells us how much trust we can place in our estimates.
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8. Disnguish between random and subjecve sampling. What is simple random sampling
? Discuss its merits and limitaons.
Ans: Random Sampling – “Chance decides, not the person”
Think about a lucky draw. Everyone puts their name chit in a box, and the teacher randomly
pulls out a few slips. Nobody can influence which name gets picked. Everyone has an equal
chance of being selected.
This is exactly what Random Sampling means.
In random sampling, the researcher does not personally select who will be included in the
sample. Instead, chance or probability decides it. Just like in a lottery, every item, person, or
unit has the same likelihood of selection.
✔ Key Idea
No human bias.
No favoritism.
Purely based on luck or chance.
Subjective Sampling – “Researcher decides based on judgment”
Now imagine another situation. Your teacher wants to know students’ views about exams,
but instead of picking names randomly, she personally chooses:
• class toppers
• a few average students
• students she knows well
• students who are easily available nearby
Here, selection depends on the teacher’s personal judgment, convenience, or opinion. This
is called Subjective Sampling (also called Non-Random Sampling).
It is called “subjective” because it is influenced by the subject (person) who is selecting.
Their thinking, experience, or assumptions play a major role.
✔ Key Idea
Selection is not equal for everyone.
Some have greater chance of being chosen than others.
Human judgment and bias can influence selection.
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Difference Between Random and Subjective Sampling (In Simple Words)
Below is a clear comparison in very easy language:
Random Sampling
Subjective Sampling
Based on chance or probability
Based on human judgment or convenience
Every member has equal chance
Only selected members have a chance
Less bias
More bias possible
More scientific and reliable
Less scientific, sometimes unreliable
Mostly used in research and statistics
Often used when time, money, or data is limited
In short:
Random sampling = fair lottery
Subjective sampling = personal choice
What is Simple Random Sampling?
Simple Random Sampling is the most basic and most important type of random sampling.
It means:
Every unit in the population has an equal and independent chance of being selected.
Let’s understand this with a relatable example.
Imagine your college has 2000 students. You want to select 200 students for a survey. You
can do any of the following:
• Assign a number to every student from 1 to 2000
• Write all numbers on slips of paper, put them in a box, and randomly pick 200
OR
• Use a computer or calculator to generate random numbers
OR
• Use lottery system
Whichever way you do it, the important point is:
Selection happens purely by chance.
No favoritism, no personal selection.
This is Simple Random Sampling.
Merits (Advantages) of Simple Random Sampling
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Simple random sampling is highly respected in statistics and research. Let us see why.
1. Free From Bias
Since every unit has equal chance, there is no favoritism.
The researcher’s personal liking or disliking cannot influence selection.
Therefore, results are more fair and trustworthy.
2. Scientific and Logical Method
This method is based on probability theory.
This makes it more scientific, systematic, and accurate compared to subjective methods.
3. Easy to Understand
The idea is very simple: “Pick randomly.”
Even a beginner can understand and apply it easily.
4. Good Representation of Population
Because selection is random, the sample generally represents the whole population quite
well.
So, conclusions drawn are usually more reliable and valid.
5. Helps in Statistical Analysis
Statistical formulas and techniques work best when random sampling is used.
So it is extremely useful in surveys, research, experiments, and social studies.
Limitations / Demerits of Simple Random Sampling
Just like everything else in life, this method also has some drawbacks.
1. Requires Complete List of Population
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You must know every member of the population and list them properly.
Without a full list, you cannot randomly select.
Example:
If you want to sample farmers in India, can you get a complete list of every single farmer?
Not always! So random sampling becomes difficult.
2. Time-Consuming and Costly
If the population is huge and spread across wide areas, selecting randomly and contacting
each sample unit may take a lot of time, money, and effort.
3. Not Always Practical
For small studies or urgent surveys, people may not have enough resources to perform
random selection. In such cases, subjective sampling becomes easier.
4. Possibility of Unrepresentative Sample
Even though it is fair, sometimes by pure chance the selected sample may accidentally not
represent the population properly.
Example:
If you randomly select students, it is possible that most selected students may come from
same academic background, simply due to chance.
Final Summary in Simple Words
Sampling helps us study a large population by selecting only a part of it.
There are two main ways:
Subjective Sampling
• Based on human judgment
• Not everyone has equal chance
• More bias
• Less scientific
Random Sampling
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• Based on chance or lottery-like method
• Everyone has equal chance
• Less bias
• More scientific
Simple Random Sampling
It is the simplest and purest form of random sampling where:
• every member has equal chance
• selection is independent
• usually done by lottery or random number method
It is highly scientific, fair, and widely used, but it requires time, money, and a complete list
of population.
“This paper has been carefully prepared for educaonal purposes. If you noce any
mistakes or have suggesons, feel free to share your feedback.”
