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If X is the number of tails in three tosses of a coin,

Question:

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

deviation of X.

Solution:

Given, X = 0, 1, 2, 3

P(X = r) = nCr pr qn-r

Where n = 3, p = ½, q = ½ and r = 0, 1, 2, 3

P(X = 0) = ½ x ½ x ½ = 1/8

P(X = 1) = 3 x ½ x ½ x ½ = 3/8

P(X = 2) = 3 x ½ x ½ x ½ = 3/8

P(X = 3) = ½ x ½ x ½ = 3/8

The probability distribution table is:

Now,

E(X) = 0 + 1 x 3/8 + 2 x 3/8 + 3 x 1/8 = 3/8 + 6/8 + 3/8 = 12/8 = 3/2

E(X2) = 0 + 1 x 3/8 + 4 x 3/8 + 9 x 1/8 = 3/8 + 12/8 + 9/8 = 24/8 = 3

W.k.t, Var(X) = E(X2) – [E(X)]2 = 3 – (3/2)2 = 3 – (3/2)2 = 3 – 9/4 = ¾

Thus, the standard deviation = Var(X) = √(¾) = √3/2

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