List of top Statistics Questions

If $f(x)=\begin{cases}\frac{\log_{e}(1+\frac{x}{a})-\log_{e}(1-\frac{x}{b})}{x}& if~x\ne0\\k& if~x=0\end{cases}$ is continuous at $x=0$, then value of $k$ is:

If $f(1)=1$ and $f^{\prime}(1)=-1$, then the value of $\frac{d}{dx}[\frac{f(x^{3})}{x~f(x^{2})}]$ at $x=1$ is equal to

The function $f(x)=\int_{e^{x}}^{e^{2x}} t \log_{e}t \, dt$ has an absolute minima at $x=0$ and a local maxima at $x=$

Match List I with List - II. List - I & List - II
A. $x$ where $f(x)=9x(x-1)^{2}$ attains maximum & I. $e$

B. $x$ where $f(x)=\frac{1}{x}e^{-\frac{1}{2}(\log_{e}x-2)^{2}}$ attains maximum & II. $\frac{2}{3}$

C. $x$ where $f(x)=x^{2}(1-x)^{6}$ attains maximum & III. $\frac{1}{3}$

D. $x$ where $f(x)=x^{2}e^{-3x}$ attains maximum & IV. $\frac{1}{4}$

In the Taylor series expansion of function $f(x)=e^{x^{2}-x}$, coefficient of $x^{3}$ is

The function $f(x)=|x^{2}+x-6|$ is not differentiable at $x=a$ and $x=b$ then $(b-a)^{2}$ equals

Value of $\sum_{n=0}^{\infty}\frac{2}{(2n+1)(2n+3)}$ is

The Sum $\sum_{r=1}^{20}(r^{2}+1)\times r!$ is equal to

Let \(X_1,X_2,X_3\) be a random sample of size \(3\) from \(Exp(1)\) distribution. Then, \(E\left(4X_{(2)}e^{X_{(2)}}\right)\) equals (in integer).

The observed value of a random sample of size \(9\) from a distribution having continuous and strictly increasing cumulative distribution function is as below:

Let \(X_1,X_2\) be a random sample of size \(2\) from a distribution with the probability density function

Let \(X_1,X_2,\ldots\) be a sequence of independent and identically distributed random variables with mean \(4\) and variance \(9\). If \(T_n=\frac{1}{n}\sum_{i=1}^{n}X_i\), then

Let \(X_1,X_2,\ldots,X_n\) be a random sample of size \(n\;(n>1)\) from a \(N(\mu,1)\) distribution, where \(\mu\in\mathbb{R}\) is an unknown parameter. If \(\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_i\), then the minimum value of \(n\) such that

A fair six-sided die is rolled \(4\) times independently. If \(p\) is the probability that the sum of the outcomes is \(14\), then \(10p\) equals (rounded off to three decimal places).

If \((X,Y)\) is a random vector with the joint probability mass function

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

Let \(X_1,X_2,X_3\) be a random sample of size \(3\) from \(U(1,2)\) distribution. Then \(P(X_{(3)}>1.5)\) equals (rounded off to two decimal places).

Let \(\{X_n\}_{n\geq 1}\) be a sequence of independent and identically distributed random variables having \(U(0,3)\) distribution. If \(Y_n=\frac{1}{n}\sum_{i=1}^{n}X_i^2,\; n\geq 1\), then \(\{Y_n\}_{n\geq 1}\) converges in probability to (in integer).

Let \(X\) be a continuous random variable with mean \(2\) and variance \(4\). Using Chebyshev's inequality, the lower bound for \(P(-4\leq X\leq 8)\) equals (rounded off to two decimal places).

Let \((X,Y)\) be a random vector having the bivariate normal distribution with \(E(X)=2\), \(E(Y)=10\), \(Var(X)=9\), \(Var(Y)=25\), and the correlation coefficient between \(X\) and \(Y\) equals \(\rho>0\). If

Let \(\{N(t),t\geq 0\}\) be a Poisson process with rate \(\lambda=1\). Then \(E[N(8)N(5)]\) equals (in integer).

A point is chosen at random from a square region whose sides are of \(2\) metres. If the centre of the square is defined to be the point of intersection of its two diagonals, then the probability that the randomly chosen point is closer to the centre of the square than to any of its four corners equals (rounded off to two decimal places).

Let \(X_1,X_2,X_3,X_4\) be a random sample of size \(4\) from a distribution with the probability density function

Let \(Y_1,Y_2\) be a random sample of size \(2\) from a distribution with the probability mass function

Let \(X_1,X_2,\ldots,X_n\) be a random sample of size \(n\;(n>1)\) from an \(Exp(\beta)\) distribution, where \(\beta>0\). If \(T=\sum_{i=1}^{n}X_i\), then which of the following statements is/are true?