The random variable x represents the number of boys in a family of three children. Assuming that boys and girls are equally likely, find the mean and standard deviation for the random variable x.
A)mean: 1.50; standard deviation: 0.87
B)mean: 2.25; standard deviation: 0.87
C)mean: 1.50; standard deviation: 0.76
D)mean: 2.25; standard deviation: 0.76

Question

Quizplus · Accepted Answer

The mean number of boys ($ \mu $) in a family of three children, given that boys and girls are equally likely, can be calculated using the formula for the mean of a binomial distribution, $ \mu = np $, where $ n = 3 $ (the number of trials, or children, in this case) and $ p = 0.5 $ (the probability of having a boy). Thus, $ \mu = 3 \times 0.5 = 1.5 $. The standard deviation ($ \sigma $) is given by the square root of the variance ($ \sigma^2 = np(1-p) $), so $ \sigma = \sqrt{3 \times 0.5 \times (1-0.5)} = \sqrt{3 \times 0.5 \times 0.5} = \sqrt{0.75} \approx 0.87 $.

The Random Variable X Represents the Number of Boys in a Family