List of top Statistics Questions on Random variables asked in CUET (PG)

The joint density function of X and Y is $f_{XY}(x,y)=\begin{cases}x+y & \text{for } 0<x<1, 0<y<1 \\ 0 & \text{elsewhere}\end{cases}$ then $P(X<2Y)$ is

Let $X_{1}, X_{2}, \dots, X_{n}$ be random sample from Normal population with mean $\mu$ and variance $\sigma^{2}$. Then which of the following results are correct?
A. $\overline{X}\sim N(\mu,\frac{\sigma^{2}}{n})$

B. $\sum_{i=1}^{n}(\frac{X_{i}-\overline{X}}{\sigma})^{2}\sim\chi_{n}^{2}$

C. $\overline{X}$ and $\sum_{i=1}^{n}(\frac{X_{i}-\overline{X}}{\sigma})^{2}$ are independently distributed

D. $\frac{(\overline{X}-\mu)^{2}}{\frac{\sigma^{2}}{n}}\sim \chi_{1}^{2}$

E. $\sum_{i=1}^{n}(\frac{X_{i}-\mu}{\sigma})^{2}\sim \chi_{n-1}^{2}$

If the random variables X and Y follows discrete uniform over set $\{0,1,...,n\}$ and $\{1,2,...,n\}$ respectively then $Var(X) - Var(Y)$ equals to

The joint density of random variable X and Y is $f_{XY}(x,y)=\begin{cases}2x & \text{for } 0<x<1, x<y<x+1 \\ 0 & \text{otherwise}\end{cases}$ then marginal of Y is

The Moment Generating Function (MGF) of random variable $X$ is given by $M_{X}(t)=(\frac{e^{-t}+e^{t}}{2})^{3}, t\ge0$ then $P(|X|>1)$ is

If $X$ and $Y$ are independent non-degenerated random variables then $Var(XY)=Var(X)\cdot Var(Y)$ iff

If $G(x)$ be the distribution function of random variable $X$ symmetric about $0$ then $\int_{-a}^{a} G(x)dx$ equals

Let $G_x(\cdot)$ be the distribution function of an arbitrary random variable symmetric about $0$ (zero) and $G_x^{\leftarrow}$ is the inverse function of $G_x$ then for $p \in (0, 1)$ value of $G_x^{\leftarrow}(p) + G_x^{\leftarrow}(1-p)$ is

If $X, X_1, X_2$ are independent and identically distributed positive random variables with distribution function $F_X(x)$ then $\int_{0}^{\infty} 2 \cdot x \cdot \overline{F}_X^2(x) dx$ equals

If $r \cdot v X \sim N(0, 1)$ then $E\left(\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{X} e^{-z^2/2} dz\right)$ equals to

In a set of 2n observations the geometric mean of first 'n' observations is 81 and the geometric mean of remaining n-observations is 16 then the geometric mean of all 2n observations is

Let $X_{1}, X_{2}$ be independent random variables each from a discrete probability mass function $P_{X}(x) = \begin{cases} 1/3 & \text{if } x = 0 \\ 2/3 & \text{if } x = 1 \end{cases}, i = 1, 2$. Then the moment generating function of $Y = X_{1} \cdot X_{2}$ is

Let X be a random variable with distribution function $F(x) = \begin{cases} 0 & \text{for } x < 0 \\ \frac{1 + x}{8} & \text{for } 0 \le x < 1 \\ \frac{x + 4}{8} & \text{for } 1 \le x < 2 \\ \frac{x + 16}{24} & \text{for } 2 \le x < 3 \\ 1 & \text{for } x \ge 3 \end{cases}$ then $P(1 \le X < 2)$ is

Let, random variable \(X \sim \text{Bernoulli}(p)\). Then, \(\beta_1\) is

Three urns contain 3 green and 2 white balls, 5 green and 6 white balls and 2 green and 4 white balls respectively. One ball is drawn at random from each of the urn. Then, the expected number of white balls drawn, is

Let \(X_1, X_2, X_3\) be three variables with means 3, 4 and 5 respectively, variances 10, 20 and 30 respectively and \(cov (X_1, X_2) = cov (X_2, X_3) = 0\) and \(cov (X_1, X_3) = 5\). If, \(Y = 2X_1 +3X_2+4X_3\) then, Var(\(Y\)) is:

Moment generating function of a random variable Y, is \( \frac{1}{3}e^t(e^t - \frac{2}{3}) \), then E(Y) is given by

If, \(f(X) = \frac{C\theta^x}{x}\); \(x = 1,2, \dots\); \(0<\theta<1\), then E(X) is equal to

If, \(f(x; \alpha, \beta) = \begin{cases} \alpha \beta x^{\beta-1} e^{-\alpha x^\beta} & ; x>0 \text{ and } \alpha, \beta>0 \\ 0 & ; \text{otherwise} \end{cases}\), then the probability density function of \(Y=x^\beta\) is

If \(f(X) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}; -\infty<x<\infty\) and \(Y = |X|\), then E(Y) is