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*0.250+1*0.441+2*0.259+3*0.050=1.109`

The standard deviation is given by
`√(\mathbb E(X^2)-\mathbb E(X)^2)`. You have


`\mathbb E(X^2)=\displaystyle\sum_xx^2p(x)=0^2*0.250+1^2*0.441+2^2*0.259+3^2*0.050=1.927`

so the standard deviation is approximately
`√(1.927-1.109^2)\approx0.835`.