GATE CS 2026 Set 1 Q48 — Probability: Random Variables and Distributions

The expected value $E[X]$ of a discrete random variable $X$ is defined as:

We are given the probability distribution of $X$ over the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$:

• For $x \in \{1, 2, 5, 7\}$, $\operatorname{Pr}(X = x) = \frac{1}{6}$
• For $x \in \{3, 4, 6, 8\}$, $\operatorname{Pr}(X = x) = \frac{1}{12}$

Let us compute the contribution of each group to the expectation:

1. Group 1: $x \in \{1, 2, 5, 7\}$ $$\sum_{x \in \{1, 2, 5, 7\}} x \cdot \operatorname{Pr}(X=x) = (1 + 2 + 5 + 7) \times \frac{1}{6} = 15 \times \frac{1}{6} = 2.5$$
2. Group 2: $x \in \{3, 4, 6, 8\}$ $$\sum_{x \in \{3, 4, 6, 8\}} x \cdot \operatorname{Pr}(X=x) = (3 + 4 + 6 + 8) \times \frac{1}{12} = 21 \times \frac{1}{12} = 1.75$$

Adding the two contributions gives the total expected value:

Thus, the expected value of $X$ is 4.25.