300+ REAL TIME Probability & Statistics Objective Questions & Answers 2023

250+ TOP MCQs on F-Distribution and Answers

Probability and Statistics Multiple Choice Questions & Answers (MCQs) on “F-Distribution”.

1. The mean of the f – distribution is equal to ___________
a) v₂ / (v₂ – 2) for v₂ > 2
b) v₂ / (v₂ – 2)² for v₂ > 2
c) v₂ / (v₂ – 2)³ for v₂ > 2
d) v₂ / (v₂ – 2)^-1 for v₂ > 2
Answer: a
Clarification: The mean of the distribution is equal to v₂ / (v₂ – 2) for v₂ > 2. v₂ denotes the degree of freedom of F-Distribution.

2. Variance is equal to [(v₁ + v₂ – 2)] / [v₁ * (v₂ – 2)² * (v₂ – 4)] for v₂ > 4 for a f-Distribution.
a) True
b) False
Answer: b
Clarification: Variance is equal to [2 * (v_2^2) * (v₁ + v₂ – 2)] / [ v₁ * (v₂ – 2)² * (v₂ – 4)] for v₂ > 4 for a f-Distribution where v₁ and v₂ denote the degrees of freedom of f-Distribution.

3. Which of the following distributions is Continuous?
a) Binomial Distribution
b) Hyper-geometric Distribution
c) F-Distribution
d) Poisson Distribution
Answer: c
Clarification: Binomial, Poisson and Hyper geometric distributions are Discrete Distributions. Only F- Distribution is Continuous Distribution in the given Distributions.

4. Which of the following distributions is used to compare two variances?
a) T – Distribution
b) F – Distribution
c) Normal Distribution
d) Poisson Distribution
Answer: b
Clarification: F – Distribution is used when we require for comparing two variances. It uses a f-Test to compare two values of variances.

5. F-Distribution cannot take negative values.
a) True
b) False
Answer: a
Clarification: The value of the F-distribution is always positive, or zero. The variances are the square of the deviations and hence cannot assume negative values. Its value lies between 0 and ∞.

6. Find Variance for an F-Distribution with v₁=5 and v₂=9.
a) 1.587
b) 1.378
c) 1.578
d) 1.498
Answer: a
Clarification: For a f – Distribution:
Var(X) = [2 * (v_2^2) * (v₁ + v₂ – 2)] / [v₁ * (v₂ – 2)² * (v₂ – 4)] for v₂ > 4 where v₁ and v₂ denote the degrees of freedom of f-Distribution.
Hence Var(X) = [2 * 9² * (14 – 2)] / [5 * (9 – 2)² * (9- 4)] = 1.587.

7. The table shows the standard Deviation and Sample Standard Deviation for both men and women. Find the f statistic considering the Men population in numerator.

Population	Population Standard Deviation	Sample Standard Deviation
Men	30	35
Women	50	45

a) 2.68
b) 1.34
c) 1.68
d) 1.34
Answer: c
Clarification: The f -statistic is calculated using the following equation:
f = [s₁²/σ₁²]/[s₂²/σ₂²].
where σ₁ is the standard deviation of population 1
s₁ is the standard deviation of the sample drawn from population 1
σ₂ is the standard deviation of population 2
s₁ is the standard deviation of the sample drawn from population 2.
f = (35²/30²)/(45²/50²)
f = (1225/900)/(2025/2500)
f = 1.361/0.81 = 1.68.

8. Calculate the value of f-statistic having a cumulative probability of 0.95.
a) 0.55
b) 0.5
c) 0.05
d) 0.05
Answer: d
Clarification: The relation between f – statistic and cumulative probability is given as
If f – statistic = f α then, cumulative probability = (1 – α)
Hence, for cumulative probability 0.95
f – statistic = (1 – 0.95) = 0.05.

9. There is only 1 parameter in F-Distribution.
a) True
b) False
Answer: b
Clarification: There are 2 parameters in F-Distribution v₁ and v₂. They are called degrees of freedom of F-Distribution.

10. Find the Expectation for a F- Distribution variable with v₁ = 7 and v₂ = 8.
a) 4/7
b) 4/6
c) 4/3
d) 4/5
Answer: c
Clarification: The Expectation for F-Distribution is given as
E(X) = v₂ / (v₂ – 2) for v₂ > 2
Hence, E(X) = 8 / (8-2)
E(X) = 4/3.

250+ TOP MCQs on Probability Distributions and Answers

Probability and Statistics Problems on “Probability Distributions – 2”.

1. If the values taken by a random variable are negative, the negative values will have ___________
a) Positive probability
b) Negative Probability
c) May have negative or positive probabilities
d) Insufficient data
Answer: a
Clarification: Probabilities are always positive and not greater than 1.

2. If f(x) is a probability density function of a continuous random variable, then (int_{-∞}^∞)f(x)=?
a) 0
b) 1
c) undefined
d) Insufficient data
Answer: b
Clarification: Sum of all probabilities of a sample space is always 1.

3. The variable that assigns a real number value to an event in a sample space is called ___________
a) Random variable
b) Defined variable
c) Uncertain variable
d) Static variable
Answer: a
Clarification: The above given statement is the definition of a random variable.

4. A random variable that assumes a finite or a countably infinite number of values is called ___________
a) Continuous random variable
b) Discrete random variable
c) Irregular random variable
d) Uncertain random variable
Answer: b
Clarification: The given statement is the definition of a discrete random variable.

5. A random variable that assume a infinite or a uncountably infinite number of values is called ___________
a) Continuous random variable
b) Discrete random variable
c) Irregular random variable
d) Uncertain random variable
Answer: a
Clarification: The given statement is the definition of a continuous random variable.

6. If Σ P(x) = k² – 8 then, the value of k is?
a) 0
b) 1
c) 3
d) Insufficient data
Answer: c
Clarification: Σ P(x) = k² – 8 = 1
On solving, we get k = 3.

7. If P(x) = 0.5 and x = 4, then E(x) = ?
a) 1
b) 0.5
c) 4
d) 2
Answer: d
Clarification: E(x) = x P(x) = 0.5 * 4 = 2.

8. In a discrete probability distribution, the sum of all probabilities is always?
a) 0
b) Infinite
c) 1
d) Undefined
Answer: c
Clarification: It is based on the basic axiom of probability distribution.

9. The expected value of a random variable is its ___________
a) Mean
b) Standard Deviation
c) Mean Deviation
d) Variance
Answer: a
Clarification: Expected value and Mean are one and the same.

10. The covariance of two independent random variable is ___________
a) 1
b) 0
c) – 1
d) Undefined
Answer: b
Clarification: Two random variables are said to be independent if their covariance is zero.

11. The weight of persons in a state is a ___________
a) Continuous random variable
b) Discrete random variable
c) Irregular random variable
d) Not a random variable
Answer: a
Clarification: Since the distribution is continuous, its a continuous random variable.

12. In random experiment, observations of random variable are classified as ___________
a) Events
b) Composition
c) Trials
d) Functions
Answer: a
Clarification: None.

250+ TOP MCQs on Testing of Hypothesis and Answers

Probability and Statistics Multiple Choice Questions & Answers (MCQs) on “Testing of Hypothesis”.

1. A statement made about a population for testing purpose is called?
a) Statistic
b) Hypothesis
c) Level of Significance
d) Test-Statistic
Answer: b
Clarification: Hypothesis is a statement made about a population in general. It is then tested and correspondingly accepted if True and rejected if False.

2. If the assumed hypothesis is tested for rejection considering it to be true is called?
a) Null Hypothesis
b) Statistical Hypothesis
c) Simple Hypothesis
d) Composite Hypothesis
Answer: a
Clarification: If the assumed hypothesis is tested for rejection considering it to be true is called Null Hypothesis. It gives the value of population parameter.

3. A statement whose validity is tested on the basis of a sample is called?
a) Null Hypothesis
b) Statistical Hypothesis
c) Simple Hypothesis
d) Composite Hypothesis
Answer: b
Clarification: In testing of Hypothesis a statement whose validity is tested on the basis of a sample is called as Statistical Hypothesis. Its validity is tested with respect to a sample.

4. A hypothesis which defines the population distribution is called?
a) Null Hypothesis
b) Statistical Hypothesis
c) Simple Hypothesis
d) Composite Hypothesis
Answer: c
Clarification: A hypothesis which defines the population distribution is called as Simple hypothesis. It specifies all parameter values.

5. If the null hypothesis is false then which of the following is accepted?
a) Null Hypothesis
b) Positive Hypothesis
c) Negative Hypothesis
d) Alternative Hypothesis.
Answer: d
Clarification: If the null hypothesis is false then Alternative Hypothesis is accepted. It is also called as Research Hypothesis.

6. The rejection probability of Null Hypothesis when it is true is called as?
a) Level of Confidence
b) Level of Significance
c) Level of Margin
d) Level of Rejection
Answer: b
Clarification: Level of Significance is defined as the probability of rejection of a True Null Hypothesis. Below this probability a Null Hypothesis is rejected.

7. The point where the Null Hypothesis gets rejected is called as?
a) Significant Value
b) Rejection Value
c) Acceptance Value
d) Critical Value
Answer: d
Clarification: The point where the Null Hypothesis gets rejected is called as Critical Value. It is also called as dividing point for separation of the regions where hypothesis is accepted and rejected.

8. If the Critical region is evenly distributed then the test is referred as?
a) Two tailed
b) One tailed
c) Three tailed
d) Zero tailed
Answer: a
Clarification: In two tailed test the Critical region is evenly distributed. One region contains the area where Null Hypothesis is accepted and another contains the area where it is rejected.

9. The type of test is defined by which of the following?
a) Null Hypothesis
b) Simple Hypothesis
c) Alternative Hypothesis
d) Composite Hypothesis
Answer: c
Clarification: Alternative Hypothesis defines whether the test is one tailed or two tailed. It is also called as Research Hypothesis.

10. Which of the following is defined as the rule or formula to test a Null Hypothesis?
a) Test statistic
b) Population statistic
c) Variance statistic
d) Null statistic
Answer: a
Clarification: Test statistic provides a basis for testing a Null Hypothesis. A test statistic is a random variable that is calculated from sample data and used in a hypothesis test.

11. Consider a hypothesis H₀ where ϕ₀ = 5 against H₁ where ϕ₁ > 5. The test is?
a) Right tailed
b) Left tailed
c) Center tailed
d) Cross tailed
Answer: a
Clarification: In the given example since H₁ lies to the right of the H_o that is the Null Hypothesis the test is referred as a Right tailed test.

12. Consider a hypothesis where H₀ where ϕ₀ = 23 against H₁ where ϕ₁ < 23. The test is?
a) Right tailed
b) Left tailed
c) Center tailed
d) Cross tailed
Answer: b
Clarification: In the Normal Distribution curve of both the hypothesis the H₁ hypothesis lies to the left of the Null hypothesis hence the test is a Left tailed.

13. Type 1 error occurs when?
a) We reject H₀ if it is True
b) We reject H₀ if it is False
c) We accept H₀ if it is True
d) We accept H₀ if it is False
Answer: a
Clarification: In Testing of Hypothesis Type 1 error occurs when we reject H₀ if it is True. On the contrary a Type 2 error occurs when we accept H₀ if it is False.

14. The probability of Type 1 error is referred as?
a) 1-α
b) β
c) α
d) 1-β
Answer: c
Clarification: In Testing of Hypothesis Type 1 error occurs when we reject H₀ if it is True. The probability of H₀ is α then the error probability will be 1- α.

15. Alternative Hypothesis is also called as?
a) Composite hypothesis
b) Research Hypothesis
c) Simple Hypothesis
d) Null Hypothesis
Answer: b
Clarification: Alternative Hypothesis is also called as Research Hypothesis. If the Null Hypothesis is false then Alternative Hypothesis is accepted.

250+ TOP MCQs on Mean and Variance of Distribution and Answers

Probability and Statistics Multiple Choice Questions & Answers (MCQs) on “Mean and Variance of Distribution”.

1. The expectation of a random variable X (E(X)) can be written as _________
a) (frac{d}{dt} [M_X (t)](t=0) )
b) (frac{d}{dx} [M_X (t)](t=0) )
c) (frac{d^2}{dt^2} [M_X (t)](t=0) )
d) (frac{d^2}{dx^2} [M_X (t)](t=0) )
Answer: a
Clarification: Expectation of a random variable X can be written as the first differentiation of Moment generating function, which can be written as (frac{d}{dt} [M_X (t)](t=0). )

2. If the probability of hitting the target is 0.4, find mean and variance.
a) 0.4, 0.24
b) 0.6, 0.24
c) 0.4, 0.16
d) 0.6, 0.16
Answer: a
Clarification: p = 0.4
q = 1-p
= 1-0.4 = 0.6
Therefore, mean = p = 0.4 and
Variance = pq = (0.4) (0.6) = 0.24.

3. If the probability that a bomb dropped from a place will strike the target is 60% and if 10 bombs are dropped, find mean and variance?
a) 0.6, 0.24
b) 6, 2.4
c) 0.4, 0.16
d) 4, 1.6
Answer: b
Clarification: Here, p = 60% = 0.6 and q = 1-p = 40% = 0.4 and n = 10
Therefore, mean = np = 6
Variance = npq = (10)(0.6)(0.4)
= 2.4.

4. If P(1) = P(3) in Poisson’s distribution, what is the mean?
a) (sqrt{2} )
b) (sqrt{3} )
c) (sqrt{6} )
d) (sqrt{7} )
Answer: c
Clarification: (P(x) = frac{(e^{-λ} λ^x)}{x!} )
Therefore, (P(3) = frac{(e^{-λ} λ^3)}{3!} )
and (P(1) = frac{(e^{-λ} λ^1)}{1!} )
P(1) = P(2)
(λ=frac{λ^3}{6} )
Therefore, (λ=sqrt{6}. )

5. What is the mean and variance for standard normal distribution?
a) Mean is 0 and variance is 1
b) Mean is 1 and variance is 0
c) Mean is 0 and variance is ∞
d) Mean is ∞ and variance is 0
Answer: a
Clarification: The mean and variance for the standard normal distribution is 0 and 1 respectively.

6. Find λ in Poisson’s distribution if the probabilities of getting a head in biased coin toss as (frac{3}{4} ) and 6 coins are tossed.
a) 3.5
b) 4.5
c) 5.5
d) 6.6
Answer: b
Clarification: p = ³⁄₄
λ = np = (6) ³⁄₄ = 4.5.

7. If P(6) = λP(1) in Poisson’s distribution, what is the mean?(Approximate value)
a) 4
b) 6
c) 5
d) 7
Answer: c
Clarification: (frac{e^{-λ} λ^6}{6!}= λ frac{e^{-λ} λ^1}{1!} )
λ⁴ = 6! = 720
Therefore λ = 5.18 = 5.

8. Find f(2) in normal distribution if mean is 0 and variance is 1.
a) 0.1468
b) 0.1568
c) 0.1668
d) 0.1768
Answer: a
Clarification: Given mean = 0
Variance = 1
(f(2) = frac{1}{(sqrt{2π})} e^{frac{-1}{2} frac{2}{1}}= 0.1468. )

9. Find the mean of tossing 8 coins.
a) 2
b) 4
c) 8
d) 1
Answer: b
Clarification: p = ¹⁄₂
n = 8
q = ¹⁄₂
Therefore, mean = np = 8 * ¹⁄₂ = 4.

10. Mean and variance of Poisson’s distribution is the same.
a) True
b) False
Answer: a
Clarification: The mean and variance of Poisson’s distribution are the same which is equal to λ.

250+ TOP MCQs on Testing of Hypothesis Concerning Single Population Mean and Answers

Probability and Statistics Quiz on “Testing of Hypothesis Concerning Single Population Mean”.

1. What is the assumption made for performing the hypothesis test with T distribution?
a) the distribution is non-symmetric
b) the distribution has more than one modal class
c) the distribution has a constant variance
d) the distribution follows a normal distribution
Answer: d
Clarification: For testing of Hypothesis with T distribution it is assumed that the distribution follows a normal distribution. The region is identified and hence based on the normal variate Hypothesis is accepted or rejected.

2. If a hypothesis is rejected at 0.6 Level of Significance then ______________
a) it will be rejected at any level
b) it must be rejected at 0.5 level
c) it may be rejected at 0.5 level
d) it cannot be rejected at 0.5 level
Answer: c
Clarification: If the hypothesis is rejected at 0.6 Level of Significance then p < 0.6. Hence p can be less than 0.5 also. Therefore it may be rejected at 0.5 Level of Significance.

3. In a two tailed test when a Null Hypothesis is rejected for a True Alternative Hypothesis then it has ____________
a) Type 1 error
b) Type 2 error
c) No error
d) Many errors
Answer: c
Clarification: In Testing of Hypothesis Type 1 error occurs when we reject a true Null Hypothesis. On the contrary a Type 2 error occurs when we accept a false Null Hypothesis. Hence if the Alternative Hypothesis is true and Null Hypothesis is rejected then no error occurs.

4. In a hypothesis test, what does the p value signify?
a) smallest level of significance for rejection of Null Hypothesis
b) largest level of significance for rejection of Null Hypothesis
c) smallest level of significance for acceptance of Null Hypothesis
d) smallest level of significance for acceptance of Null Hypothesis
Answer: a
Clarification: In a Hypothesis, the p value signifies the smallest level of significance for rejection of Null Hypothesis. Below this value, for every value the hypothesis is rejected.

5. A Null Hypothesis has Level of Significance 9%. For what values of Level of Significances it will be rejected?
a) 0.99
b) 0.009
c) 0.099
d) 0.9
Answer: b
Clarification: The Level of Significance of Null Hypothesis is 0.09. Hence the hypothesis will be rejected at values less than 0.09.

6. Consider a trial of a criminal. If a type 1 error has occurred in thee judgement then which of the following statement is true?
a) a guilty person is set free
b) an innocent person is convicted
c) a guilty person is convicted
d) an innocent person is set free
Answer: b
Clarification: Type 1 error occurs when the Null hypothesis is True but rejected. Considering Null hypothesis as innocent person and setting free as acceptance an innocent person getting convicted is a type 1 error.

7. If a Null Hypothesis is accepted then the value of Test statistic lies in the ____________
a) Acceptance region
b) Rejection region
c) Critical region
d) Sample region
Answer: a
Clarification: If a Null Hypothesis is accepted then the value of Test statistic lies in the Acceptance region. For a rejected Null Hypothesis the value lies in the Rejection region.

8. The Test Statistic for a Hypothesis testing is given by the formula ____________
a) Sample-Population/Standard Error
b) Sample statistic-Parameter/Standard Error
c) Sample mean-Population mean/Population standard deviation
d) Statistic-E(statistic)/Variance
Answer: b
Clarification: The Test Statistic of a hypothesis is given as the Sample statistic-Parameter/Standard Error.
The Test statistic provides a basis for testing a Null Hypothesis.

9. The range of Level of Significance lies between ____________
a) -∞ and 0
b) -∞ and ∞
c) 0 and ∞
d) 0 and 1
Answer: d
Clarification: The Level of Significance lies between 0 and 1. The 0 signifies the test is least significant and 1 signifies the test is most significant.

10. The effect of rejection of a hypothesis with decrease in sample size ____________
a) decreases
b) increases
c) remains constant
d) fluctuates
Answer: b
Clarification: If n decreases then the value of Level of Significance of each sample α decreases. Hence 1-α increases which are called the rejection of test sample increases.

11. The composite hypothesis holds true when?
a) ϕ > ϕ₀
b) ϕ < ϕ₀
c) ϕ = ϕ₀
d) ϕ >> ϕ₀
Answer: c
Clarification: The composite hypothesis holds true when the Null Hypothesis is equal to the Alternative Hypothesis. Hence ϕ = ϕ₀. Here the value of α is 0.5.

12. A paired T test consists of n pairs of observations. What is the number of degrees of freedom of the test?
a) 2n-1
b) 2n
c) n-1
d) n
Answer: c
Clarification: For a paired T distribution the no of degrees of Freedom are n-1 where n denotes the number of pairs of samples in the test.

13. Which of the following represents the Confidence coefficient?
a) 1-α
b) β
c) 1-β
d) α
Answer: a
Clarification: The level of Confidence is represented by 1-α. It signifies the chance of the Alternative Hypothesis.

14. The independent values in a set of values of a test is called as?
a) Degrees of freedom
b) Test Statistic
c) Level of Significance
d) Level of Confidence
Answer: a
Clarification: In a test, the number of individual samples is called as Degrees of Freedom. If a sample has n values then the Degrees of Freedom are n-1.

15. A T-test sample has 7 pairs of samples. The distribution should contain ____________
a) 16 degrees of freedom
b) 15 degrees of freedom
c) 5 degrees of freedom
d) 6 degrees of freedom
Answer: d
Clarification: Here the number of samples, n is 7. Hence total degrees of Freedom is n-1 that is 7-1 = 6.

250+ TOP MCQs on Mathematical Expectation and Answers

Probability and Statistics Multiple Choice Questions & Answers (MCQs) on “Mathematical Expectation”.

1. The expectation of a random variable X(continuous or discrete) is given by _________
a) ∑xf(x), ∫xf(x)
b) ∑x² f(x), ∫x² f(x)
c) ∑f(x), ∫f(x)
d) ∑xf(x²), ∫xf(x²)
Answer: a
Clarification: The expectation of a random variable X is given by the summation (integral) of x times the function in its interval. If it is a continuous random variable, then summation is used and if it is discrete random variable, then integral is used.

2. Mean of a random variable X is given by _________
a) E(X)
b) E(X²)
c) E(X²) – (E(X))²
d) (E(X))²
Answer: a
Clarification: Mean is defined as the sum of the function in its domain multiplied with the random variable’s value. Hence mean is given by E(X) where X is a random variable.

3. Variance of a random variable X is given by _________
a) E(X)
b) E(X²)
c) E(X²) – (E(X))²
d) (E(X))²
Answer: c
Clarification: Variance of a random variable is nothing but the expectation of the square of the random variable subtracted by the expectation of X (mean of X) to the power 2. Therefore the variance is given by E(X²) – (E(X))².

4. Mean of a constant ‘a’ is ___________
a) 0
b) a
c) a/2
d) 1
Answer: b
Clarification: Let f(x) be the pdf of the random variable X.
Now, E(a) = ∫af(x)
= a∫f(x)
= a(1) = a.

5. Variance of a constant ‘a’ is _________
a) 0
b) a
c) a/2
d) 1
Answer: a
Clarification: V(a) = E(a²) – (E(X))²
= a² – a²
= 0.

6. Find the mean and variance of X?

x	0	1	2	3	4
f(x)	1/9	2/9	3/9	2/9	1/9

a) 2, 4/3
b) 3, 4/3
c) 2, 2/3
d) 3, 2/3
Answer: a
Clarification: Mean = (E(X) = ∑f(x) = 0(frac{1}{9}) + 1(frac{2}{9}) + 2(frac{3}{9}) + 3(frac{2}{9}) + 4(1/9) )
= 2
Variance ( = E(X^2)-(E(X))^2 = (0 + frac{2}{9} + frac{12}{9} + frac{28}{9} + frac{26}{9}) – 4 )
( = frac{4}{3} ).

7. Find the expectation of a random variable X?

x	0	1	2	3
f(x)	1/6	2/6	2/6	1/6

a) 0.5
b) 1.5
c) 2.5
d) 3.5
Answer: b
Clarification: (E(X) = 0(frac{1}{6}) + 1(frac{2}{6}) + 2(frac{2}{6}) + 3(frac{1}{6}) = 1.5. )

8. Find the expectation of a random variable X if f(x) = ke^-x for x>0 and 0 otherwise.
a) 0
b) 1
c) 2
d) 3
Answer: b
Clarification: (int_0^∞ ke^{-x} dx = 1 )
kГ(1) = 1
k = 1
Now, (E(X) = int_0^∞ xe^{-x} dx = Г(2) = 1.)

9. Find the mean of a random variable X if f(x) = x – ⁵⁄₂ for 0a) 3.5
b) 3.75
c) 2.5
d) 2.75
Answer: b
Clarification: (E(X) = int_0^1 (x-5/2)dx+∫_1^2(2x)dx+0 )
(= (frac{x^3}{3} – frac{5x^2}{4}) ) {from 0 to 1} ( + (frac{2x^3}{3}) ) {from 1 to 2}
(= frac{1}{3} – frac{5}{4} + frac{16}{3} – frac{2}{3} )
= 3.75.

10. Find the mean of a continuous random variable X if f(x) = 2e^-x for x>0 and -e^x for x<0.
a) 0
b) 1
c) 2
d) 3
Answer: d
Clarification: (E(X) = int_0^∞ 2xe^{-x} dx + int_{-∞}^0 xe^x dx )
= 2 Г(2) + Г(2) = 3.

11. What is moment generating function?
a) M_x(t) = E(e^tx)
b) M_x(t) = E(e^-tx)
c) M_x(t) = E(e^2tx)
d) M_x(t) = E(e^t)
Answer: a
Clarification: Moment generating function is nothing but the expectation of e^tX. So, the function is multiplied with e^tX before performing the integration or summation.

12. Find the Moment Generating Function of f(x) = x for 0a) ((frac{e^t-1}{t})^2 )
b) ((frac{e^{-t}-1}{t})^2 )
c) ((frac{e^{2t}-1}{t})^2 )
d) ((frac{e^{2t}-1}{t^2}) )
Answer: a
Clarification: M_x(t) = E(e^tx) = (int_0^1 xe^{tx} dx+int_1^2 (2-x) e^{tx} dx + 0 = (frac{e^t-1}{t})^2. )

13. E(X) = npq is for which distribution?
a) Bernoulli’s
b) Binomial
c) Poisson’s
d) Normal
Answer: b
Clarification: In binomial distribution, probability of success is given by p and that of failure is given by q and the event is done n times. The mean of this distribution is given by npq.

14. E(X) = λ is for which distribution?
a) Bernoulli’s
b) Binomial
c) Poisson’s
d) Normal
Answer: c
Clarification: In Poisson’s distribution, there is a positive constant λ which is the mean of the distribution and variance of the distribution.

15. E(X) = μ and V(X) = σ² is for which distribution?
a) Bernoulli’s
b) Binomial
c) Poisson’s
d) Normal
Answer: d
Clarification: In Normal distribution, the mean and variance is given by μ and σ² respectively. In case of standard normal distribution the mean is 0 and the variance is 1.