Let X be a random variable which takes values in the set {1, 2, 3, 4, 5, 6, 7, 8}. Further, (X = 1) = (X = 2) = (X = 5) = (X = 7) = (1)/(6) and (X = 3) = (X = 4) = (X = 6) = (X = 8) = (1)/(12). The expected value of X, denoted by E[X], is equal to … (rounded off to two decimal places)

Official answer: 4.25 (±0.01). Explanation: The expected value E[X] of a discrete random variable X is defined as: E[X] = Σx x · (X = x) Partition the values according to their given probabilities: 1. For values x ∈ {1, 2, 5, 7}, each has probability (1)/(6): Σx ∈ {1, 2, 5, 7} x · (1)/(6) = (1 + 2 + 5 + 7) × (1)/(6) = 15 × (1)/(6) = (15)/(6) = 2.5 2. For values x ∈ {3, 4, 6, 8}, each has probability (1)/(12): Σx ∈ {3, 4, 6, 8} x · (1)/(12) = (3 + 4 + 6 + 8) × (1)/(12) = 21 × (1)/(12) = (21)/(12) = 1.75 Summing these contributions: E[X] = 2.5 + 1.75 = 4.25 Thus, E[X] = 4.25.

Probability and Statistics

GATE CSE 2026 Set 1 · Q58

NAT2 marksEasy

Time00:00

Let $X$ be a random variable which takes values in the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ . Further, $\Pr(X = 1) = \Pr(X = 2) = \Pr(X = 5) = \Pr(X = 7) = \frac{1}{6}$ and $\Pr(X = 3) = \Pr(X = 4) = \Pr(X = 6) = \Pr(X = 8) = \frac{1}{12}$ . The expected value of $X$ , denoted by $E[X]$ , is equal to ________. (rounded off to two decimal places)

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Solution

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Official answer: 4.25 (±0.01) (±0.01)

Probability and Statistics

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

E[X] = \sum_{x} x \cdot \Pr(X = x)

Partition the values according to their given probabilities:

For values $x \in \{1, 2, 5, 7\}$ , each has probability $\frac{1}{6}$ :

\sum_{x \in \{1, 2, 5, 7\}} x \cdot \frac{1}{6} = (1 + 2 + 5 + 7) \times \frac{1}{6} = 15 \times \frac{1}{6} = \frac{15}{6} = 2.5

For values $x \in \{3, 4, 6, 8\}$ , each has probability $\frac{1}{12}$ :

\sum_{x \in \{3, 4, 6, 8\}} x \cdot \frac{1}{12} = (3 + 4 + 6 + 8) \times \frac{1}{12} = 21 \times \frac{1}{12} = \frac{21}{12} = 1.75

Summing these contributions:

E[X] = 2.5 + 1.75 = 4.25

Thus, $E[X] = 4.25$ .

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This Probability and Statistics previous year question is Q58 of the GATE CSE 2026 Set 1 paper. Find more GATE CSE PYQs on Probability and Statistics in Engineering Mathematics, or browse all GATE CSE PYQs.

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Probability and Statistics has 46 GATE CSE PYQs across 23 papers from 2011 to 2026. 15 of them are 2-mark NAT questions like this one, and GATE CSE 2026 Set 1 has 2.

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.