Question

Let the random variable x represent the number of girls in a family with three children. Assume the probability of a child being a girl is 0.37. The table on the right describes the probability of having x number of girls. Determine whether the table describes a probability distribution. If it​ does, find the mean and standard deviation. Is it unusual for a family of three children to consist of three​ girls?
x p(x)
0 .250
1 .441
2 .259
3 .050

Answer 1

All the probabilities add up to 1, so the table does indeed describe a distribution.

The mean is given by

$\mathbb E(X)=\displaystyle\sum_x xp(x)=0\times0.250+1\times0.441+2\times0.259+3\times0.050=1.109$

The standard deviation is given by $\sqrt{\mathbb E(X^2)-\mathbb E(X)^2}$. You have

$\mathbb E(X^2)=\displaystyle\sum_xx^2p(x)=0^2\times0.250+1^2\times0.441+2^2\times0.259+3^2\times0.050=1.927$

so the standard deviation is approximately $\sqrt{1.927-1.109^2}\approx0.835$.