GATE CS 2025 Set 2 Q55 — Probability: Random Variables and Distributions

Let the division point be represented by the random variable $X$, which is uniformly distributed over the interval $(0, 1)$. The probability density function of $X$ is:

The point $X$ divides the interval $(0, 1)$ into two subintervals: $(0, X)$ and $(X, 1)$.

The subinterval containing $0.4$ depends on the position of $X$:

• If $X \le 0.4$, the subinterval containing $0.4$ is $(X, 1)$, which has a length of $1 - X$.
• If $X > 0.4$, the subinterval containing $0.4$ is $(0, X)$, which has a length of $X$.

Thus, the length of the subinterval containing $0.4$, denoted by $L(X)$, is defined piecewise as:

The expected length $\mathbb{E}[L(X)]$ is calculated as follows:

Evaluating the two integrals separately:

Adding the two parts together: