Find \(E(X)\) and \(V(X)\), where X is the number obtained on uppermost face, when a fair die is thrown.

Q.25 Find \(E(X)\) and \(V(X)\), where X is the number obtained on uppermost face, when a fair die is thrown.

Doubtly Asked 28 Oct 2026 Board exam · March 2026 hsc 12th Save WhatsApp Report

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Answer: \(E(X) = \dfrac{7}{2} = 3.5\), \(V(X) = \dfrac{35}{12} \approx 2.917\)

\(X\) takes the values 1, 2, 3, 4, 5, 6, each with probability \(\dfrac{1}{6}\).

\(x\)	1	2	3	4	5	6
\(P(x)\)	\(\tfrac{1}{6}\)	\(\tfrac{1}{6}\)	\(\tfrac{1}{6}\)	\(\tfrac{1}{6}\)	\(\tfrac{1}{6}\)	\(\tfrac{1}{6}\)
\(x^2\)	1	4	9	16	25	36

\(E(X) = \sum xP(x) = \dfrac{1 + 2 + 3 + 4 + 5 + 6}{6} = \dfrac{21}{6} = \dfrac{7}{2}\).

\(E(X^2) = \sum x^2P(x) = \dfrac{1 + 4 + 9 + 16 + 25 + 36}{6} = \dfrac{91}{6}\).

\(V(X) = E(X^2) - [E(X)]^2 = \dfrac{91}{6} - \dfrac{49}{4} = \dfrac{182 - 147}{12} = \dfrac{35}{12}\).

Doubtly AI · 28 Oct 2026 AI-verified · Oct 2026 Report

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