List of top Statistics Questions asked in CUET (PG)

Let $X_{1}, X_{2}, \dots$ be independent variables each taking values $+1$ or $-1$ with equal probability respectively. If $S_{n}=\sum_{i=1}^{n} i X_{i}$ then $\lim_{n \rightarrow \infty} P\left(S_{n} < \sqrt{\frac{n(n+1)(2 n+1)}{3}}\right)$ where $\Phi$ is distribution function of standard normal variate, is

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

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

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

If $X$ has the $F$ distribution with $m, n$ degree of freedoms and let $Y=\frac{1}{X}$ then for $a>0$ $P[X\le a]+P[Y\le\frac{1}{a}]$ is equal to

Let $X_{1}$ and $X_{2}$ be i.i.
D. Bernoulli (p), $0

If $X_{1}, X_{2}, X_{3}$ are independent and identically distributed standard normal variates and let $U=\frac{\sqrt{2}X_{3}}{\sqrt{X_{1}^{2}+X_{2}^{2}}}$ then $U^{2}$ follows

A fair coin is tossed $2n$ times, then the probability that the outcomes do not result in an equal number of heads and tails is

The variance of random variable $X$ having density $f_{X}(x)=ce^{-|x|}, -\infty<x<\infty$ is

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

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

Let $X$ and $Y$ be independent non negative integer valued random variables with $E(X) < \infty$, $E(Y) < \infty$, then

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

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

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

You are given $P(A \cup B) = 0.6$ and $P(A \cup \overline{B}) = 0.8$ then $P(A)$ 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

If G is a geometric mean of observations $x_{1}, x_{2}, \dots, x_{n}$ then the geometric mean of $y_{i} = e^{-\alpha \log_{e} x_{i}}$, $i = 1, 2, \dots, n$ is

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 E, F and G be mutually independent events such that $P(E) = 0.4$, $P(F) = 0.6$ and $P(G) = 0.8$ then $P(\overline{E} \cup \overline{F} \cup G)$ is

Consider $x_{1},x_{2},...,x_{n}$ observations such that $\sum_{i=1}^{n}{x_{i}}^{2}=500$ and $\sum_{i=1}^{n}x_{i}=50$. Then a minimum number of observations required is

Three dice have the probabilities of throwing a "five" as p, q and r respectively. One of the dice is chosen at random (each is equally likely to be chosen) and thrown and a "five" appeare
D. What is the probability that the die chosen was the first one?

Let E, F and G be events such that $P(E|G) = 0.05$ and $P(F|G) = 0.05$ which of the following statement must be true?