Question

If X is the number of tails in three tosses of a coin, determine the standard deviation of X.

Answer 1

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`.