GATE EC 2025 Set 1 Q48 — 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 pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement. What is the probability that the two balls drawn have different colours?

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?

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).