List of top Statistics Questions

Let $X$ be a random variable having probability density function $f_{X}(x)=\begin{cases}2x & 0<x<1 \\ 0 & \text{otherwise}\end{cases}$ then the density of $Y=\frac{1}{X^{\frac{1}{\alpha}}}$ 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

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

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

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

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

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

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

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

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

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

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

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 $X_{1}, X_{2}$ 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

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

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

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

If $P(E) = \frac{1}{3}$, $P(F) = \frac{2}{5}$ and $P(E \cup F) - P(E \cap F) = \frac{1}{5}$ then $P(E \cup F)$ is equal to

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?

Which of the following statements are correct?
A. Ogive curves are used to obtain median

B. Histogram are used to obtain mode

C. Boxplots are used to determine mean

D. Pie charts are used to determine quantile

Choose the correct answer from the options given below