Consider the 3-child, equally likely genders, experiment. Let Xi be the number o

Question

Consider the 3-child, equally likely genders, experiment. Let Xi be the number of girls on the ith birth (Xi= 0 or 1 for i = 1,2 or 3).

a) Show that X = X1 + X2 + X3 is the random variable indication the total number of girls in the three child family. Note: X is called a binomial random variable.

b) calculate the mean and the variance of Xi. (Since the Xi]s are identically distributed you need do the calculation only once)

c) Use the pair-wise independence of the Xi's and formulas (a) and (c) of the theorem to calculate the mean and the variance of X.

Explanation / Answer

(a)

X=X1+X2+X3 where

Xi = 1 if a girl is born in ith birth

= 0 otherwise , i=1,2,3

The distribution of Xi is P[Xi =1] = P[Xi = 0] = 1/2

Xi ~ Bin( 1 , 1/2) i.i.d.

=> X ~ Bin(3,1/2)

Here X is the total number of girls out of 3 children born.

(b)

E(Xi) = 0*1/2 + 1*1/2 = 1/2 , i=1,2,3

E(Xi2) = 02*1/2 + 12*1/2 = 1/2 , i=1,2,3

V(Xi) = E(Xi2) - E2(Xi) = 1/2 - 1/4 = 1/4 , i=1,2,3

(c)

E(X) =E(X1+X2+X3 ) = E(X1)+E(X2)+E(X3) = 1/2 + 1/2 +1/2 = 3/2

V(X) = V(X1+X2+X3 ) = V(X1)+V(X2)+V(X3) = 1/4 + 1/4 +1/4 = 3/4 (covariance terms do not appear as Xi s are pairwise independent)

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