2025 Ststistics Paper - I

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

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?

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

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?

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?

The distribution function of standard Laplace distribution with the pdf f(x) = 1/2 e^{-|x|}, -\infty < x < \infty is :