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Question
If X is the number of tails in three tosses of a coin, determine the standard deviation of X.
Solution
Given that: X = 0, 1, 2, 3
∴ P(X = r) = `""^"n""C"_"r" "p"^"r" "q"^("n" - "r")`
Where n = 3
p = `1/2`
q = `1/2`
And r = 0, 1, 2, 3
P(X = 0) = `1/2 xx 1/2 xx 1/2 = 1/8`
P(X = 1) = `3 xx 1/2 xx 1/2 xx 1/2 = 3/8`
P(X = 2) = `3 xx 1/2 xx 1/2 xx 1/2 = 3/8`
P(X = 3) = `1/2 xx 1/2 xx 1/2 = 1/8`
Probability distribution table is:
| X | 0 | 1 | 2 | 3 |
| P(X) | `1/8` | `3/8` | `3/8` | `1/8` |
E(X) = `0 + 1 xx 3/8 + 2 xx 3/8 + 3 xx 1/8`
= `3/8 + 6/8 + 3/8`
= `12/8`
= `3/2`
E(X2) = `0 + 1 xx 3/8 + 4 xx 3/8 + 9 xx 1/8`
= `3/8 + 12/8 + 9/8`
= `24/8`
= 3
We know that Var(X) = E(X2) – [E(X)]2
= `3 - (3/2)^2`
= `3 - 9/4`
= `3/4`
∴ Standard deviation = `sqrt("Var"("X"))`
= `sqrt(3/4)`
= `sqrt(3)/2`.
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