GATE EC 2025 Set 1 Q3 — Probability and Statistics

Let $X, N, Y$ and $Z$ be random variables. The variables $X$ and $N$ are independent of each other. $X$ is uniformly distributed between $-1$ and $1 ; N$ follows Normal distribution with zero mean and unity variance. $Y$ and $Z$ are defined as, $Y=X+N$ and $Z=X^{2}+N$. Which of the following pairs represents the values of correlation …

A source transmits symbol $S$ that takes values uniformly at random from the set $\{-2,0,2\}$. The receiver obtains $Y=S+N$, where $N$ is a zero-mean Gaussian random variable independent of $S$. The receiver uses the maximum likelihood decoder to estimate the transmitted symbol $S$. Suppose the probability of symbol estimation error $P_{e}$ is …

The random variable $X$ takes values in $\{-1,0,1\}$ with probabilities $P(X=-1)=P(X=1)=\alpha$ and $P(X=0)=1-2 \alpha$, where $0<\alpha<1 / 2$. Let $g(\alpha)$ denote the entropy of $X$ (in bits), parameterized by $\alpha$. Which of the following statements is/are TRUE?

Two fair dice (with faces labeled $1,2,3,4,5$, and $6$) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained. The expectation of $X$ is __________ (rounded off to two decimal places).

Suppose $\text{X}$ and $\text{Y}$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability that $\text{X} \geq \text{Y}$ is $\_\_\_\_\_\_$.

A random variable $\mathrm{X}$, distributed normally as $\mathrm{N(0,1)}$ undergoes the transformation $\mathrm{Y}=\mathrm{h}(\mathrm{X})$, given in the figure. The form of the probability density function of $\mathrm{Y}$ is (In the options given below, $a, b, c$ are non-zero constants and $g(y)$ is piece-wise continuous function).