Distribution MCQ Quiz - Objective Question with Answer for Distribution - Download Free PDF

Last updated on Jun 24, 2025

Latest Distribution MCQ Objective Questions

Distribution Question 1:

Let X1, X2,X3 be a random sample from a continuous distribution having cumulative distribution function F(t), probability density function f(t), and failure rate function r(t)=f(t)1F(t),t>0, where F(0) = 0. If r(t) = 1 for all t >.0, then which of the following statements are true? 

  1. P(max{X1,x2}<1)=12e
  2. P(min{X1,X2}>1)=12e
  3. \(\rm P(min\{X_1, X_2\}
  4. \(\rm P(max\{X_1, X_2\}

Answer (Detailed Solution Below)

Option :

Distribution Question 1 Detailed Solution

Concept:

For an exponential distribution with parameter  λ=1 :
 

  F(t)=1et is the CDF.


f(t)=et is the PDF.

Explanation:

Option 1:The maximum of two independent exponential random variables X1 and X2 follows the formula

P(max{X1,X2}<1)=F(1)2=(1e1)2=(11e)2.

and  (11e)2=121e+1e2=12e+1e2.

This expression is not equal to 12e  , so Option 1 is false.

Option 2: The minimum of two independent exponential random variables  X1 and X2 follows the formula:

P(min{X1,X2}>1)=(1F(1))2=e1e1=e2.

This is not equal to 12e , so Option 2 is false.

Option 3: For three independent exponential random variables, we need to compute the probability that the minimum

of  X1 and X2 is greater than X3 . The joint distribution and comparison of exponential variables leads to this result

being known as a standard property of exponential distributions. The probability for this scenario is 23 , so Option 3 is true.

Option 4: Similarly, using properties of the exponential distribution and the comparison of order statistics, the probability

that the maximum of   X1 and X2   is less than X3is known to be 13, so Option 4 is true.

Hence, option 3) and 4) are correct.

Distribution Question 2:

Let X1....Xn (n ≥ 3) be a random sample from a distribution having probability density function f(|x|θ)={θeθx,if x>0 0,otherwise

where θ > 0 is an unknown parameter. Let Tn=1nΣi=1nXi Which of the following statements are true? 

  1. Uniformly minimum variance unbiased estimator of θ is n1nTn
  2. Cramer-Rao lower bound for the variance of any unbiased estimator of θ is θ2n
  3. Uniformly minimum variance unbiased estimator of θ attains the Cramer-Rao lower bound 
  4. (1e1Tn) is a consistent estimator of pθ (X1 ≤ 1)

Answer (Detailed Solution Below)

Option :

Distribution Question 2 Detailed Solution

Concept:

Probability Density Function (PDF) for Exponential Distribution

The probability density function (PDF) of an exponential distribution is given by:

f(x|θ)={θeθxif x>00otherwise

Here, θ is the rate parameter. The exponential distribution is often used to model time

until the next event in processes where events occur continuously and independently

at a constant average rate (e.g., Poisson process).

From the sample X1,X2,,Xn, the sample mean Tn=1ni=1nXi  is a sufficient statistic for θ.

The sample mean Tn  is an estimator for 1θ  because for an exponential distribution, the expected value is E(Xi)=1θ.

Explanation: The problem you are dealing with involves a random sample X1,X2,,Xn from a distribution with the given probability density function (PDF):

f(x|θ)={θeθxif x>00otherwise

where θ>0 is an unknown parameter, and the statistic Tn=1ni=1nXi is considered.

Option 1: For an exponential distribution, the sample mean Tn=1ni=1nXi is a sufficient and

unbiased estimator of 1θ. Therefore, the unbiased estimator of θ is 1Tn.

The UMVUE of θ is n1nTn, as this is a correction that adjusts for bias. So, option 1 is true.

 Option 2: The Fisher information I(θ) for a single observation from the exponential distribution θeθx is 1θ2.

Therefore, for a sample of size n, the Fisher information is n×1θ2 , and the Cramer-Rao lower bound

(CRLB) for the variance of any unbiased estimator of θ is:

CRLB=1n×I(θ)=θ2n.

So, option 2 is true.

Option 3: The UMVUE of θ does not necessarily attain the Cramer-Rao lower bound. While it is unbiased

and has the minimum variance among unbiased estimators, it might not reach the bound. Option 3 is false.

Option 4: The probability Pθ(X11) can be calculated from the cumulative distribution function

(CDF) of the exponential distribution:

Pθ(X11)=1eθ.

Using the fact that Tn is an unbiased estimator for 1θ, the expression (1e1Tn) is a consistent

estimator for 1eθ, which is Pθ(X11). Therefore, option 4 is true.

The correct statements are Option 1), Option 2) and Option 4).

Distribution Question 3:

Let a point P be chosen at random on the line segment AB of length α. Let Z1 and Z2 denote the lengths of line segments AP and BP respectively. Then the value of E(|Z1 - Z2|) Is 

  1. α 
  2. 2α 
  3. α2
  4. 2α3

Answer (Detailed Solution Below)

Option 3 : α2

Distribution Question 3 Detailed Solution

Concept:

Expected Value Calculation:

The expected value E(|Z1Z2|)  can be derived using geometric probability or integrating over

the length of the segment. For a uniform distribution of the point P on the segment, the expected value of

the absolute difference between the two distances is known to be

E(|Z1Z2|)=α2

Explanation:

Let us introduce the variable X=Z1Z2. Since Z1+Z2=α, we have  Z2=αZ1

Thus, X=Z1(αZ1)=2Z1α , so  X=2Z1α .

We want to compute E(|X|), which is equivalent to E(|2Z1α|).

Distribution of  Z1

Since  P  is chosen uniformly along the line segment Z1, is uniformly distributed

between  0 and  α. The probability density function (PDF) of Z1 is

  fZ1(z1)=1αfor0z1α

Expected Value of |X|

Now, we compute the expected value E(|2Z1α|). This involves integrating the

absolute value of 2Z1α over the range of Z1:


E(|2Z1α|)=0α|2z1α|1αdz1

Splitting the Integral

The expression |2Z1α| changes sign when z1=α2.

So, we split the integral into two parts:

For 0z1α2, |2z1α|=α2z1.

For α2z1α, |2z1α|=2z1α.

Therefore, the expected value becomes:

 E(|2Z1α|)=0α2(α2z1)1αdz1+α2α(2z1α)1αdz1   

First integral 0α2(α2z1)1αdz1=1α[αz1z12]0α2=1α(α22α24)=α4

Second integral α2α(2z1α)1αdz1=1α[z12αz1]α2α=1α(α2α2+α24)=α4
 
 E(|2Z1α|)=α4+α4=α2
 
E(|Z1Z2|)=α2

Hence option 3) is correct.
 

Distribution Question 4:

Consider a system with two components whose lifetimes are i.i.d. exponential with hazard rate λ. Let h1 and h2 be the hazard functions of the system if the components are put in series and parallel, respectively. Then which of the following are true?

  1. h2(t) < h1(t) for all t > 0
  2. h2(t) < λ for all t > 0
  3. h1(t) < λ for all t > 0
  4. h2 is a strictly increasing function of t

Answer (Detailed Solution Below)

Option :

Distribution Question 4 Detailed Solution

The correct answer is options are 1, 2 & 4. 

we will update the solution as soon as possible.

Distribution Question 5:

Suppose a normal Q - Q plot is drawn using a reasonably large sample x1,... xn from an unknown probability distribution. For which of the following distributions would you expect the Q - Q plot to be convex (J - shaped)?

  1. Beta (5, 1)
  2. Exponential (1)
  3. Uniform (0, 1) 
  4. Geometric (1/2)

Answer (Detailed Solution Below)

Option :

Distribution Question 5 Detailed Solution

The correct answer is options are 2 & 4. 

we will update the solution as soon as possible.

Top Distribution MCQ Objective Questions

Let a point P be chosen at random on the line segment AB of length α. Let Z1 and Z2 denote the lengths of line segments AP and BP respectively. Then the value of E(|Z1 - Z2|) Is 

  1. α 
  2. 2α 
  3. α2
  4. 2α3

Answer (Detailed Solution Below)

Option 3 : α2

Distribution Question 6 Detailed Solution

Download Solution PDF

Concept:

Expected Value Calculation:

The expected value E(|Z1Z2|)  can be derived using geometric probability or integrating over

the length of the segment. For a uniform distribution of the point P on the segment, the expected value of

the absolute difference between the two distances is known to be

E(|Z1Z2|)=α2

Explanation:

Let us introduce the variable X=Z1Z2. Since Z1+Z2=α, we have  Z2=αZ1

Thus, X=Z1(αZ1)=2Z1α , so  X=2Z1α .

We want to compute E(|X|), which is equivalent to E(|2Z1α|).

Distribution of  Z1

Since  P  is chosen uniformly along the line segment Z1, is uniformly distributed

between  0 and  α. The probability density function (PDF) of Z1 is

  fZ1(z1)=1αfor0z1α

Expected Value of |X|

Now, we compute the expected value E(|2Z1α|). This involves integrating the

absolute value of 2Z1α over the range of Z1:


E(|2Z1α|)=0α|2z1α|1αdz1

Splitting the Integral

The expression |2Z1α| changes sign when z1=α2.

So, we split the integral into two parts:

For 0z1α2, |2z1α|=α2z1.

For α2z1α, |2z1α|=2z1α.

Therefore, the expected value becomes:

 E(|2Z1α|)=0α2(α2z1)1αdz1+α2α(2z1α)1αdz1   

First integral 0α2(α2z1)1αdz1=1α[αz1z12]0α2=1α(α22α24)=α4

Second integral α2α(2z1α)1αdz1=1α[z12αz1]α2α=1α(α2α2+α24)=α4
 
 E(|2Z1α|)=α4+α4=α2
 
E(|Z1Z2|)=α2

Hence option 3) is correct.
 

Let X1, X2, ..., Xn be i.i.d. random variables with common pdf

f(xθ)=(logθ)θxθ1, for

0 < x < 1 where θ > 1 is an unknown parameter. Then the statistic T=i=1nXi is

  1. sufficient, but not complete
  2. Sufficient, but not minimal sufficient
  3. Complete sufficient
  4. neither complete, nor sufficient

Answer (Detailed Solution Below)

Option 3 : Complete sufficient

Distribution Question 7 Detailed Solution

Download Solution PDF

Given:-

X1, X2, ..., Xn are i.i.d. random variables with common pdf

f(xθ)=(logθ)θxθ1

Concept Used:-

To check whether T = ∑ᵢ Xᵢ is a sufficient statistic for the parameter θ, we need to find the likelihood function of the sample.

Explanation:-

The joint pdf of the sample is given by

f(x,x,...,x|θ)=f(x|θ)f(x|θ)...f(x|θ)f(x,x,...,x|θ)=i=1nf(xiθ)f(x,x,...,x|θ)=i=1n(logθ)θxiθ1 f(x,x,...,x|θ)=(logθ)nθi=1nxi(θ1)n

The factorization theorem states that a statistic T is sufficient for a parameter θ if and only if the joint pdf of the sample can be factorized into two functions.

One which depends on the sample only through T, and the other that does not depend on θ.

We can rewrite the joint pdf of the sample as

f(x,x,...,x|θ)=h(x,x,...,x)g(T|θ),

Here,

h(x,x,...,x)=(logθ)n   andg(T|θ)=(θT)/(θ1)n  

Since the joint pdf can be factorized in this way, T is a sufficient statistic for θ.

To check whether T is complete, we need to show that T is able to capture all the information about θ contained in the sample.

In other words, any function of T that is independent of θ must have an expected value equal to 0.

Let h(T) be a function of T that is independent of θ. Then

E[h(T)]=h(T)g(T|θ)dTE[h(T)]=01h(T)θT(θ1)ndTE[h(T)]=(logθ)n(θ1)n01h(T)exp(Tlogθ)dT

Since θ > 1, \log θ > 0, and the integrand is positive on [0,1].

Therefore, if E[h(T)] = 0 for all θ > 1, then h(T) must be identically zero almost everywhere on [0,1].

This shows that T is a complete, sufficient statistic for θ.

Therefore, the statistic T=i=1nXi is completely sufficient.

Hence, the correct option is 3.

Distribution Question 8:

Suppose X ~ Poisson (34). Then which of the following statements is true? 

  1. P(X>9)1112
  2. P(X<9)1112
  3. E(X34)21112
  4. 119XPoisson(1112)

Answer (Detailed Solution Below)

Option 2 : P(X<9)1112

Distribution Question 8 Detailed Solution

The correct answer is option 2

we will update the solution as soon as possible.

Distribution Question 9:

Let a point P be chosen at random on the line segment AB of length α. Let Z1 and Z2 denote the lengths of line segments AP and BP respectively. Then the value of E(|Z1 - Z2|) Is 

  1. α 
  2. 2α 
  3. α2
  4. 2α3

Answer (Detailed Solution Below)

Option 3 : α2

Distribution Question 9 Detailed Solution

Concept:

Expected Value Calculation:

The expected value E(|Z1Z2|)  can be derived using geometric probability or integrating over

the length of the segment. For a uniform distribution of the point P on the segment, the expected value of

the absolute difference between the two distances is known to be

E(|Z1Z2|)=α2

Explanation:

Let us introduce the variable X=Z1Z2. Since Z1+Z2=α, we have  Z2=αZ1

Thus, X=Z1(αZ1)=2Z1α , so  X=2Z1α .

We want to compute E(|X|), which is equivalent to E(|2Z1α|).

Distribution of  Z1

Since  P  is chosen uniformly along the line segment Z1, is uniformly distributed

between  0 and  α. The probability density function (PDF) of Z1 is

  fZ1(z1)=1αfor0z1α

Expected Value of |X|

Now, we compute the expected value E(|2Z1α|). This involves integrating the

absolute value of 2Z1α over the range of Z1:


E(|2Z1α|)=0α|2z1α|1αdz1

Splitting the Integral

The expression |2Z1α| changes sign when z1=α2.

So, we split the integral into two parts:

For 0z1α2, |2z1α|=α2z1.

For α2z1α, |2z1α|=2z1α.

Therefore, the expected value becomes:

 E(|2Z1α|)=0α2(α2z1)1αdz1+α2α(2z1α)1αdz1   

First integral 0α2(α2z1)1αdz1=1α[αz1z12]0α2=1α(α22α24)=α4

Second integral α2α(2z1α)1αdz1=1α[z12αz1]α2α=1α(α2α2+α24)=α4
 
 E(|2Z1α|)=α4+α4=α2
 
E(|Z1Z2|)=α2

Hence option 3) is correct.
 

Distribution Question 10:

Consider a system with two components whose lifetimes are i.i.d. exponential with hazard rate λ. Let h1 and h2 be the hazard functions of the system if the components are put in series and parallel, respectively. Then which of the following are true?

  1. h2(t) < h1(t) for all t > 0
  2. h2(t) < λ for all t > 0
  3. h1(t) < λ for all t > 0
  4. h2 is a strictly increasing function of t

Answer (Detailed Solution Below)

Option :

Distribution Question 10 Detailed Solution

The correct answer is options are 1, 2 & 4. 

we will update the solution as soon as possible.

Distribution Question 11:

Suppose X1, X2, .., Xn is a random sample from uniform distribution on (θ, θ + 1), where θ ∈ ℝ is an unknown parameter. Let X(1) < X(2) < ... < X(n) be the corresponding order-statistics. Which of the following are 100(1 - α)% confidence intervals for θ? 

  1. (-∞, X(n) - α1/n
  2. (X(1) + α1/n - 1, ∞) 
  3. (Xn+α21,Xnα2)
  4. (-∞, X- α)

Answer (Detailed Solution Below)

Option :

Distribution Question 11 Detailed Solution

The Correct Answer is Option 1, 2, 3 & 4

Distribution Question 12:

Let X1, X2, ..., Xn be i.i.d. random variables with common pdf

f(xθ)=(logθ)θxθ1, for

0 < x < 1 where θ > 1 is an unknown parameter. Then the statistic T=i=1nXi is

  1. sufficient, but not complete
  2. Sufficient, but not minimal sufficient
  3. Complete sufficient
  4. neither complete, nor sufficient

Answer (Detailed Solution Below)

Option 3 : Complete sufficient

Distribution Question 12 Detailed Solution

Given:-

X1, X2, ..., Xn are i.i.d. random variables with common pdf

f(xθ)=(logθ)θxθ1

Concept Used:-

To check whether T = ∑ᵢ Xᵢ is a sufficient statistic for the parameter θ, we need to find the likelihood function of the sample.

Explanation:-

The joint pdf of the sample is given by

f(x,x,...,x|θ)=f(x|θ)f(x|θ)...f(x|θ)f(x,x,...,x|θ)=i=1nf(xiθ)f(x,x,...,x|θ)=i=1n(logθ)θxiθ1 f(x,x,...,x|θ)=(logθ)nθi=1nxi(θ1)n

The factorization theorem states that a statistic T is sufficient for a parameter θ if and only if the joint pdf of the sample can be factorized into two functions.

One which depends on the sample only through T, and the other that does not depend on θ.

We can rewrite the joint pdf of the sample as

f(x,x,...,x|θ)=h(x,x,...,x)g(T|θ),

Here,

h(x,x,...,x)=(logθ)n   andg(T|θ)=(θT)/(θ1)n  

Since the joint pdf can be factorized in this way, T is a sufficient statistic for θ.

To check whether T is complete, we need to show that T is able to capture all the information about θ contained in the sample.

In other words, any function of T that is independent of θ must have an expected value equal to 0.

Let h(T) be a function of T that is independent of θ. Then

E[h(T)]=h(T)g(T|θ)dTE[h(T)]=01h(T)θT(θ1)ndTE[h(T)]=(logθ)n(θ1)n01h(T)exp(Tlogθ)dT

Since θ > 1, \log θ > 0, and the integrand is positive on [0,1].

Therefore, if E[h(T)] = 0 for all θ > 1, then h(T) must be identically zero almost everywhere on [0,1].

This shows that T is a complete, sufficient statistic for θ.

Therefore, the statistic T=i=1nXi is completely sufficient.

Hence, the correct option is 3.

Distribution Question 13:

Let X1, X2,X3 be a random sample from a continuous distribution having cumulative distribution function F(t), probability density function f(t), and failure rate function r(t)=f(t)1F(t),t>0, where F(0) = 0. If r(t) = 1 for all t >.0, then which of the following statements are true? 

  1. P(max{X1,x2}<1)=12e
  2. P(min{X1,X2}>1)=12e
  3. \(\rm P(min\{X_1, X_2\}
  4. \(\rm P(max\{X_1, X_2\}

Answer (Detailed Solution Below)

Option :

Distribution Question 13 Detailed Solution

Concept:

For an exponential distribution with parameter  λ=1 :
 

  F(t)=1et is the CDF.


f(t)=et is the PDF.

Explanation:

Option 1:The maximum of two independent exponential random variables X1 and X2 follows the formula

P(max{X1,X2}<1)=F(1)2=(1e1)2=(11e)2.

and  (11e)2=121e+1e2=12e+1e2.

This expression is not equal to 12e  , so Option 1 is false.

Option 2: The minimum of two independent exponential random variables  X1 and X2 follows the formula:

P(min{X1,X2}>1)=(1F(1))2=e1e1=e2.

This is not equal to 12e , so Option 2 is false.

Option 3: For three independent exponential random variables, we need to compute the probability that the minimum

of  X1 and X2 is greater than X3 . The joint distribution and comparison of exponential variables leads to this result

being known as a standard property of exponential distributions. The probability for this scenario is 23 , so Option 3 is true.

Option 4: Similarly, using properties of the exponential distribution and the comparison of order statistics, the probability

that the maximum of   X1 and X2   is less than X3is known to be 13, so Option 4 is true.

Hence, option 3) and 4) are correct.

Distribution Question 14:

Let X1....Xn (n ≥ 3) be a random sample from a distribution having probability density function f(|x|θ)={θeθx,if x>0 0,otherwise

where θ > 0 is an unknown parameter. Let Tn=1nΣi=1nXi Which of the following statements are true? 

  1. Uniformly minimum variance unbiased estimator of θ is n1nTn
  2. Cramer-Rao lower bound for the variance of any unbiased estimator of θ is θ2n
  3. Uniformly minimum variance unbiased estimator of θ attains the Cramer-Rao lower bound 
  4. (1e1Tn) is a consistent estimator of pθ (X1 ≤ 1)

Answer (Detailed Solution Below)

Option :

Distribution Question 14 Detailed Solution

Concept:

Probability Density Function (PDF) for Exponential Distribution

The probability density function (PDF) of an exponential distribution is given by:

f(x|θ)={θeθxif x>00otherwise

Here, θ is the rate parameter. The exponential distribution is often used to model time

until the next event in processes where events occur continuously and independently

at a constant average rate (e.g., Poisson process).

From the sample X1,X2,,Xn, the sample mean Tn=1ni=1nXi  is a sufficient statistic for θ.

The sample mean Tn  is an estimator for 1θ  because for an exponential distribution, the expected value is E(Xi)=1θ.

Explanation: The problem you are dealing with involves a random sample X1,X2,,Xn from a distribution with the given probability density function (PDF):

f(x|θ)={θeθxif x>00otherwise

where θ>0 is an unknown parameter, and the statistic Tn=1ni=1nXi is considered.

Option 1: For an exponential distribution, the sample mean Tn=1ni=1nXi is a sufficient and

unbiased estimator of 1θ. Therefore, the unbiased estimator of θ is 1Tn.

The UMVUE of θ is n1nTn, as this is a correction that adjusts for bias. So, option 1 is true.

 Option 2: The Fisher information I(θ) for a single observation from the exponential distribution θeθx is 1θ2.

Therefore, for a sample of size n, the Fisher information is n×1θ2 , and the Cramer-Rao lower bound

(CRLB) for the variance of any unbiased estimator of θ is:

CRLB=1n×I(θ)=θ2n.

So, option 2 is true.

Option 3: The UMVUE of θ does not necessarily attain the Cramer-Rao lower bound. While it is unbiased

and has the minimum variance among unbiased estimators, it might not reach the bound. Option 3 is false.

Option 4: The probability Pθ(X11) can be calculated from the cumulative distribution function

(CDF) of the exponential distribution:

Pθ(X11)=1eθ.

Using the fact that Tn is an unbiased estimator for 1θ, the expression (1e1Tn) is a consistent

estimator for 1eθ, which is Pθ(X11). Therefore, option 4 is true.

The correct statements are Option 1), Option 2) and Option 4).

Distribution Question 15:

Suppose a normal Q - Q plot is drawn using a reasonably large sample x1,... xn from an unknown probability distribution. For which of the following distributions would you expect the Q - Q plot to be convex (J - shaped)?

  1. Beta (5, 1)
  2. Exponential (1)
  3. Uniform (0, 1) 
  4. Geometric (1/2)

Answer (Detailed Solution Below)

Option :

Distribution Question 15 Detailed Solution

The correct answer is options are 2 & 4. 

we will update the solution as soon as possible.

Get Free Access Now
Hot Links: teen patti master golden india teen patti master plus teen patti casino