List of top Mathematics Questions on Probability theory asked in IIT JAM EN

Let $X_1, X_2, X_3, X_4$ be independent random variables following the standard normal distribution. Let $Y$ be defined as \[ Y = (X_1 + X_2)^2 + (X_3 + X_4)^2. \] Then the variance of $Y$ equals __________. (in integer) 
 

The amount of money a gambler can win in a casino is determined by three independent rolls of a six-faced fair dice. The gambler wins Rs. 800 if he gets three sixes, Rs. 400 if he gets two sixes, and Rs. 100 in the event of getting only one six. The gambler does not win or lose any money in all other possible outcomes. The probability that a gambler will win at least Rs. 400 is ___________. (round off to 2 decimal places) 
 

Let $X$ and $Y$ be two random variables with the joint probability density function \[ f_{X,Y}(x,y) = \begin{cases} 6xy, & 0 < y \le \sqrt{x} \le 1, \\ 0, & \text{otherwise.} \end{cases} \] Then, the conditional probability $P(Y \ge \tfrac{1}{3} \mid X = \tfrac{2}{3})$ is
Let $X$ and $Y$ be two independent random variables with the cumulative distribution functions \[ F_X(x) = 1 - \left(\frac{3}{4}\right)^x, \quad x = 1,2,3,\ldots \] \[ F_Y(y) = 1 - \left(\frac{2}{3}\right)^y, \quad y = 1,2,3,\ldots \] respectively. Let $Z = \min\{X, Y\}$. Then, the probability $P(Z \ge 6)$ is
Let $X$ be a uniformly distributed random variable in $[0, b]$. If the critical region for testing the null hypothesis $H_0: b = 2$ against the alternative $H_A: b \ne 2$ is $\{x \le 0.1 \text{ or } x \ge 1.9\}$, where $x$ is the value of a single draw of $X$, then the probability of Type-I error is