Find the standard deviation of the random variable.?

Find the standard deviation of the random variable.


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. 0.984





d. 1.053





e. 1.173

Find the standard deviation of the random variable.?
Let X be the number of children. The possible values for X and the corresponding probabilities are





x...............P(X = x)





1 (B)............1/2


2 (GB).........1/4


3 (GGB)......1/8


4 (GGGB).. 2/16 = 1/8


or (GGGG)





The SD(X) is the square root of the variance of X (Var(X)). The variance is equal to





Var(X) = E(X^2) - (E(X))^2





So we need E(X) and E(X^2)





E(X) = 1*(1/2) + 2*(1/4) + 3*(1/8) + 4*(1/8) = 15/8





E(X^2) = 1*(1/2) + 4*(1/4) + 9*(1/8) + 16*(1/8) = 37/8





Var(X) = 37/8 - (15/8)^2 = 71/64





SD(X) = sqrt(71/64) = 1.053. So, answer (D)!





Math (and Stats) Rule!
Reply:Let X be the number of children. The possible values for X and the corresponding probabilities are





x...............P(X = x)





1 (B)............1/2


2 (GB).........1/4


3 (GGB)......1/8


4 (GGGB).. 2/16 = 1/8


or (GGGG)





The SD(X) is the square root of the variance of X (Var(X)). The variance is equal to





Var(X) = E(X^2) - (E(X))^2





So we need E(X) and E(X^2)





E(X) = 1*(1/2) + 2*(1/4) + 3*(1/8) + 4*(1/8) = 15/8





E(X^2) = 1*(1/2) + 4*(1/4) + 9*(1/8) + 16*(1/8) = 37/8





Var(X) = 37/8 - (15/8)^2 = 71/64





SD(X) = sqrt(71/64) = 1.053. So, answer (D)!