A couple plans to have children until they get a boy,but they agree that they will not have more than four children even if all are girls.Find the standard deviation of the number of children the couple have.Assume that boys and girls are equally likely.Round your answer to three decimal places.
A)1.109
B)0.992
C)1.053
D)0.984
E)1.173

Question

Quizplus · Accepted Answer

Let X be the number of children the couple have until they get a boy. Then X follows a geometric distribution with p = 1/2 (assuming boys and girls are equally likely). The mean of X is given by E(X) = 1/p = 2, and the variance is given by Var(X) = (1-p)/p^2 = 2.

However, we also have the condition that the couple will not have more than four children, even if all are girls. In other words, X is truncated at 4. We can adjust for this by using the truncated geometric distribution. Using the formula for the standard deviation of a truncated geometric distribution, we get:

SD(X) = sqrt{ Var(X) - [1-F(4)]/p^2 } = sqrt{2 - [1-1/16]/(1/4)^2 } = 1.053 (rounded to three decimal places)

Therefore, the answer is C, 1.053.

A Couple Plans to Have Children Until They Get a Boy,but