the random variable x represents the number of boys in a family of three children. assuming that boys and…

Explanation:

Step1: Calculate the mean

The formula for the mean (\mu) of a discrete random variable is (\mu=\sum_{x}x\cdot P(x)). [ \begin{align*} \mu&=(0\times0.125)+(1\times0.375)+(2\times0.375)+(3\times0.125)\ &=0 + 0.375+0.75 + 0.375\ &=1.5 \end{align*} ]

Step2: Calculate the variance

The formula for the variance (\sigma^{2}=\sum_{x}(x - \mu)^{2}\cdot P(x)) [ \begin{align*} \sigma^{2}&=(0 - 1.5)^{2}\times0.125+(1 - 1.5)^{2}\times0.375+(2 - 1.5)^{2}\times0.375+(3 - 1.5)^{2}\times0.125\ &=(2.25\times0.125)+(0.25\times0.375)+(0.25\times0.375)+(2.25\times0.125)\ &=0.28125+0.09375 + 0.09375+0.28125\ &=0.75 \end{align*} ]

Step3: Calculate the standard deviation

The standard deviation (\sigma=\sqrt{\sigma^{2}}) [ \sigma=\sqrt{0.75}\approx0.87 ]

Answer:

A. mean: 1.50; standard deviation: 0.87