Discrete Random Variables and Bernoulli-Binomial Models Previous Year Questions (PYQs) for GATE DA: 4+ Solved Questions with Step-by-Step Solutions

A bag contains $5$ white balls and $10$ black balls. In a random experiment, $n$ balls are drawn from the bag one at a time with replacement. Let $S_n$ denote the total number of black balls drawn in the experiment.
The expectation of $S_{100}$ denoted by $E[S_{100}]$ =
(Round off to one decimal place)

Let $A_{5\times 5}$ be a matrix such that each of its elements follows $Bernoulli(p=0.50)$ distribution independently.

The probability that the row-sum of the second row and the column-sum of the third column are both equal to 3 is ________ . (Rounded off to two decimal places)

A random variable $X$ is said to be distributed as Bernoulli$(\theta )$, denoted by $X\sim Bernoulli(\theta )$, if

$$P(X=1)=\theta ,P(X=0)=1-\theta$$
for $0<\theta <1$. Let $Y={\sum }_{i=1}^{300}{X}_i$, where $X_i\sim Bernoulli(\theta )$, $i=1,2,...,300$ be independent and identically distributed random variables with $\theta =0.25$. The value of $P(60\le Y\le 90)$, after approximation through Central Limit Theorem, is given by

Recall that $\varphi (x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^x{e}^{-\frac{t^2}{2}}dt$