NIMCET 2026 — Mathematics PYQ

Given that $X$ and $Y$ are independent Bernoulli random variables with $P(X=1)=P(Y=1)=\frac{1}{2}$, we need to find the distribution of $Z=XY$. Since $X$ and $Y$ only take values 0 or 1, their product $Z$ will also take values 0 or 1.

Step 1: Calculate $P(Z=1)$

The product $XY=1$ if and only if both $X=1$ and $Y=1$.

$$P(Z=1)=P(X=1\cap Y=1)$$

Because $X$ and $Y$ are independent:

$$P(Z=1)=P(X=1)\cdot P(Y=1)$$

$$P(Z=1)=\left(\frac{1}{2}\right)\cdot \left(\frac{1}{2}\right)=\frac{1}{4}$$

Step 2: Calculate $P(Z=0)$

Since $Z$ is a Bernoulli random variable, $P(Z=0)=1-P(Z=1)$:

$$P(Z=0)=1-\frac{1}{4}=\frac{3}{4}$$

Thus, $Z$ follows a Bernoulli distribution with parameter $p=\frac{1}{4}$.

Correct Option: (c) Bernoulli with $P(Z=1)=\frac{1}{4}$ and $P(Z=0)=\frac{3}{4}$