ISRO EC 2025 Q.18: solution and answer

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

1. A random variable $X\ge 0$ is memoryless if $P(X>s+t\mid X>s)=P(X>t)$ for all $s,t\ge 0$.

2. For the exponential distribution with rate $\lambda$, $P(X>x)=e^{-\lambda x}$ for $x\ge 0$.

3. Then $P(X>s+t\mid X>s)=\frac{P(X>s+t)}{P(X>s)}=\frac{e^{-\lambda (s+t)}}{e^{-\lambda s}}=e^{-\lambda t}=P(X>t)$, so the property holds.

4. The normal distribution is not memoryless: the conditional distribution given $X>s$ is a truncated normal, and its tail depends on $s$ (the tail ratio $P(X>s+t)/P(X>s)$ shrinks faster as $s$ grows, since the hazard rate increases).

5. The log-normal distribution is not memoryless either: the ratio $P(X>s+t)/P(X>s)$ changes with $s$, so it is not equal to $P(X>t)$.

6. Among continuous distributions the exponential is the only memoryless one, so option (c) is correct and (d) is wrong.