List of top Statistics Questions on Standard Distributions asked in CUET (PG)

If \( A_{x:n} = \{k : |k - \tfrac{n}{2}| \le \tfrac{x\sqrt{n}}{2} \} \), then the value of

\( \lim_{n \to \infty} \sum_{k \in A_{1:n}} \binom{n}{k} 2^{-n} \)

(where \( \Phi(\cdot) \) is the distribution function of the standard normal variate) is:

Let $X_{i} \sim N(0,1), i=1,2, \dots$ be independent random variables. If $T_{n}=\sum_{i=1}^{n} X_{i}^{2}$, then $\lim_{n \rightarrow \infty} P(T_{n} > n+2 \sqrt{2 n})$ is equal to

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

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

Which of the following statement is true about the geometric series \(1+r+r^2+r^3+.............; (r>0)\) ?

Out of 800 families with 4 children each, the percentage of families having no girls is:

If, joint distribution function of two random variables X and Y is given by \(F_{X,Y}(x,y) = \begin{cases} 1 - e^{-x} - e^{-y} + e^{-(x+y)} & ; x>0; y>0 \\ 0 & ; \text{otherwise} \end{cases}\), then Var(\(X\)) is

In a binomial distribution consisting of five independent trails, the probability of 1 and 2 success are 0.4096 and 0.2048 respectively. Then, the parameter 'p' of distribution is

Let, \(X \sim \beta_1(u, v)\) and \(Y \sim \gamma(1, u+v)\); (\(u, v>0\)) be independent random variables. If, \(Z = XY\), then the moment generating function of Z is given by

If X and Y are independent and identically distributed geometric variables with parameter p, then the moment generating function of (X+Y) is given by

Let, X and Y be independent and identically distributed Poisson(1) variables. If, Z = min(X, Y) then, P(Z = 1) is:

If, \(X \sim \text{Bin}(8, 1/2)\) and \(Y = X^2+2\), then \(P(Y \le 6)\) is:

If, \(f(x, y) = xe^{-x(y+1)}; x \ge 0, y \ge 0\), then \(E(Y|X = x)\) is

For two random variables X and Y having the joint probability density function \( f(x,y) = \frac{1}{3}(x+y); 0 \le x \le 1, 0 \le y \le 2 \), then cov(X, Y) is

From the data relating to the yield of dry bark (\(x_1\)), height (\(x_2\)) and girth (\(x_3\)) for 18 cinchona plants, the correlation coefficient are obtained as \(r_{12}=0.77, r_{13} = 0.72, r_{23} = 0.52\). Then, the multiple correlation coefficient \(R_{1.23}\) is

If all the zero order correlation coefficients in a set of n-variates are equal to \(\rho\), then every third order partial correlation coefficient is equal to:

In measuring reaction times, a psychologist estimates that the standard deviation is 0.05 seconds. How large a sample of measurements should be taken in order to be 95% confident that the error of the estimate will not exceed 0.01 seconds?