IES ISS 2025 Paper-I Statistics Question Paper (21-June-2025)

Question 1

X follows Beta distribution of first kind Beta (α, β) with α>1 and β>1. What is the mode?

Accepted Answer

$\frac{\alpha - 1}{\alpha + \beta - 2}$

Answer

$\frac{\alpha}{\alpha + \beta}$

Answer

$\frac{\alpha + 1}{\alpha + \beta + 1}$

Answer

$\frac{\alpha + \beta}{\alpha + \beta - 1}$

Question 2

For Beta distribution of first kind Beta (α, β), consider the following statements :
I. For α>1, β>1; density has unique mode.
II. For α + β = 2 such that α = β = 1, all points in [0, 2] are modes.
Which of the statements given above is/are correct?

Accepted Answer

I only

Answer

II only

Answer

Both I and II

Answer

Neither I nor II

Question 3

Let X be a continuous random variable with the following pdf : $f(x) = \begin{cases} \frac{1}{8}, & 	ext{for } 2 \leq x \leq 10 \ 0, & 	ext{otherwise} \end{cases}$What is the distribution of Y =  aX + b, provided a>0 ?

Accepted Answer

U(2a + b, 10a + b)

Answer

U(8a + b, 12a + b)

Answer

Poisson (ba¹ᐟ⁸)

Answer

None of the above

Question 4

A player rolls a fair dice and simultaneously tosses a fair coin. If the "head" appears on the coin, then the player wins twice the value that appears on the dice. However if the “tail” appears on the coin, then the player wins half of the value that appears on the dice. What is the player's expected win?

Accepted Answer

4.375

Answer

10.25

Answer

8.625

Answer

7.625

Question 5

A miner is trapped in a mine containing 3 doors. The first door leads to a tunnel that will take him to safety after 3 hours of travel. The second door leads to a tunnel that will return him to the mine after 5 hours of travel. The third door leads to a tunnel that will return him to the mine after 7 hours of travel. Assuming that the miner is equally likely to choose any of the doors, what is the expected length of time until he reaches safety?

Accepted Answer

15 hours

Answer

25 hours

Answer

30 hours

Answer

35 hours