Three coins are tossed simultaneously, \(X\) is the number of heads. Find the expected value and variance of \(X\).
Q.24 Three coins are tossed simultaneously, \(X\) is the number of heads. Find the expected value and variance of \(X\).
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Answer: \(E(X) = \dfrac32\), \(\operatorname{Var}(X) = \dfrac34\)
Sample space has 8 equally likely outcomes. \(X\) takes values 0, 1, 2, 3.
| \(x\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \(P(x)\) | \(\frac18\) | \(\frac38\) | \(\frac38\) | \(\frac18\) |
| \(x\,P(x)\) | 0 | \(\frac38\) | \(\frac68\) | \(\frac38\) |
| \(x^2P(x)\) | 0 | \(\frac38\) | \(\frac{12}{8}\) | \(\frac98\) |
\(E(X) = \sum xP(x) = \dfrac{12}{8} = \dfrac32\).
\(E(X^2) = \sum x^2P(x) = \dfrac{24}{8} = 3\).
\(\operatorname{Var}(X) = E(X^2) - [E(X)]^2 = 3 - \dfrac94 = \dfrac34\).
Doubtly AI
· 28 Oct 2026
AI-verified · Oct 2026
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