ISRO EC 2025 Q.86: solution and answer

Official answer: BSource: ISRO's final answer key of 15 May 2026

1. A discrete random variable takes countable values, so it is described by a probability mass function (written as a sum of impulses if a density is wanted). A continuous random variable takes values over an interval and has a genuine density function.

2. Gaussian: $f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-(x-\mu {)}^2/2{\sigma }^2}$ is defined for every real $x$, so the variable is continuous.

3. Poisson: $P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$ for $k=0,1,2,\dots$ takes only non-negative integer values, so it is discrete.

4. Rayleigh: $f(x)=\frac{x}{{\sigma }^2}{e}^{-x^2/2{\sigma }^2}$ for $x\ge 0$ is a continuous density.

5. Exponential: $f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$ is a continuous density.

6. Only the Poisson distribution describes a discrete random variable, so option (b) is correct.