List of top Probability Questions

Let $X\sim \text{Unif}(0,4)$ and, given $X=x$, $Y\mid X=x\sim \text{Unif}\!\left(0,\dfrac{x^{2}}{4}\right)$. Compute $E(Y^{2})$ (rounded off to three decimal places).

Let $Y_1<\cdots<Y_n$ be order statistics from a continuous distribution symmetric about its mean $\mu$. Find the smallest $n$ (in integer) such that $P(Y_1<\mu<Y_n)\ge 0.99$.

Suppose a random sample of size 3 is taken from a distribution with the probability density function \[ f(x) = \begin{cases} 2x, & 0<x<1, \\ 0, & \text{elsewhere}. \end{cases} \] If \( p \) is the probability that the largest sample observation is at least twice the smallest sample observation, then the value of \( p \) (rounded off to three decimal places) is ________

Let \( X \) be a random variable such that \[ P \left( \frac{a}{2\pi} X \in \mathbb{Z} \right) = 1, \quad a>0, \] where \( \mathbb{Z} \) denotes the set of all integers. If \( \phi_X(t), t \in \mathbb{R} \), denotes the characteristic function of \( X \), then which of the following is/are true?

Suppose \( X_1, X_2, \dots, X_n, \dots \) are independent exponential random variables with the mean \( \frac{1}{2} \). Let the notation \( i.o. \) denote "infinitely often." Then which of the following is/are true?

Consider the following statements:
(I) Let a random variable \(X\) have the probability density function \[ f_X(x) = \frac{1}{2} e^{-|x|}, \quad -\infty<x<\infty. \] Then there exist i.i.d. random variables \(X_1\) and \(X_2\) such that \(X\) and \(X_1 - X_2\) have the same distribution.
(II) Let a random variable \(Y\) have the probability density function \[ f_Y(y) = \begin{cases} \frac{1}{4}, & -2<y<2, \\ 0, & \text{elsewhere.} \end{cases} \] Then there exist i.i.d. random variables \(Y_1\) and \(Y_2\) such that \(Y\) and \(Y_1 - Y_2\) have the same distribution.
Then which of the above statements is/are true?

Let \( \{ X_n \}, n \geq 1 \), be a sequence of random variables with the probability mass functions \[ p_{X_n}(x) = \begin{cases} \frac{n}{n+1}, & x = 0 \\ \frac{1}{n+1}, & x = n \\ 0, & \text{elsewhere} \end{cases} \] Let \( X \) be a random variable with \( P(X = 0) = 1 \). Then which of the following statements is/are true?

Let \( X \) be a random variable with the probability density function \[ f(x) = \begin{cases} c(x - \lfloor x \rfloor), & 0<x<3, \\ 0, & \text{elsewhere}, \end{cases} \] where \( c \) is a constant and \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \). If \[ A = \left[ \frac{1}{2}, 2 \right], \text{ then } P(X \in A) \, \text{(rounded off to two decimal places) is equal to} \, ________. \]

Let \( X \) and \( Y \) be two random variables such that the moment generating function of \( X \) is \( M(t) \) and the moment generating function of \( Y \) is \[ H(t) = \left( \frac{3}{4} e^{2t} + \frac{1}{4} \right) M(t), \] where \( t \in (-h, h), h>0 \). If the mean and the variance of \( X \) are \( \frac{1}{2} \) and \( \frac{1}{4} \), respectively, then the variance of \( Y \) (in integer) is equal to ________

Let \( X_i, i = 1, 2, \dots, n \), be i.i.d. random variables with the probability density function \[ f_X(x) = \begin{cases} \frac{1}{\sqrt{2 \Gamma\left( \frac{1}{6} \right)}} x^{-\frac{5}{6}} e^{-\frac{x}{8}}, & 0<x<\infty, \\ 0, & \text{elsewhere}, \end{cases} \] where \( \Gamma(\cdot) \) denotes the gamma function. Also, let \( \bar{X}_n = \frac{1}{n} (X_1 + X_2 + \cdots + X_n) \). If \[ \sqrt{n} \left( \bar{X}_n - \mathbb{E}[\bar{X}_n] \right) \xrightarrow{d} N(0, \sigma^2), \] then \( \sigma^2 \) (rounded off to two decimal places) is equal to ________

Let \( X \) and \( Y \) be two independent exponential random variables with \( E(X^2) = \frac{1}{2} \) and \( E(Y^2) = \frac{2}{9} \). Then \( P(X<2Y) \) (rounded off to two decimal places) is equal to ________

Let \( X_i, i = 1, 2, \dots, n \), be i.i.d. random variables from a normal distribution with mean 1 and variance 4. Let \( S_n = X_1^2 + X_2^2 + \dots + X_n^2 \). If \( \text{Var}(S_n) \) denotes the variance of \( S_n \), then the value of \[ \lim_{n \to \infty} \left( \frac{\text{Var}(S_n)}{n} - \left( \frac{E(S_n)}{n} \right)^2 \right) \quad \text{(in integer) is equal to} \, ________. \]

Let \( X_1, X_2, \dots, X_{10} \) be a random sample from a probability density function

\[ f_\theta(x) = f(x - \theta), \quad -\infty<x<\infty, \]

where \( -\infty<\theta<\infty \) and \( f(-x) = f(x) \) for \( -\infty<x<\infty \). For testing

\[ H_0: \theta = 1.2 \quad \text{against} \quad H_1: \theta \neq 1.2, \]

let \( T^+ \) denote the Wilcoxon Signed-rank test statistic. If \( \eta \) denotes the probability of the event \( \{T^+<50\} \) under \( H_0 \), then \( 32\eta \) equals

\[ \underline{\hspace{2cm}} \]

(round off to 2 decimal places).

Let \( (X, Y) \) have the joint probability density function

\[ f_{X,Y}(x,y) = \begin{cases} \frac{4}{(x + y)^3}, & x>1, y>1 \\ 0, & \text{otherwise} \end{cases} \]

Then which one of the following statements is NOT true?

Let \( X \) be a random variable having the probability density function

\[ f(x) = \begin{cases} \frac{3}{13} (1 - x)(9 - x), & 0<x<1, \\ 0, & \text{otherwise}. \end{cases} \]

Then

\[ \frac{4}{3} E[(X^2 - 15X + 27)] \]

equals _________ (round off to 2 decimal places).

Consider an amusement park where visitors are arriving according to a Poisson process with rate 1. Upon arrival, a visitor spends a random amount of time in the park and then departs. The time spent by the visitors are independent of one another, as well as of the arrival process, and have common probability density function \[ f(x) = \begin{cases} e^{-x}, & x>0,
0, & \text{otherwise}. \end{cases} \] If at a given time point, there are 10 visitors in the park and \( p \) is the probability that there will be exactly two more visitors before the next departure, then \[ p = _________ \text{ (round off to 2 decimal places).} \]

Let customers arrive at a departmental store according to a Poisson process with rate 10. Further, suppose that each arriving customer is either a male or a female with probability \( \frac{1}{2} \) each, independent of all other arrivals. Let \( N(t) \) denote the total number of customers who have arrived by time \( t \). Then which one of the following statements is NOT true?