A family has four children. If Y is random variable that pertains to the number of female children what are the possible values of Y?

This is a binomial distribution with $n=4,p=\frac{1}{2}.$

$P(Y=y)=C_4^2(\frac{1}{2})^x(\frac{1}{2})^{4-x}.$

$P(Y=0)=C_4^0(\frac{1}{2})^0(\frac{1}{2})^4=\frac{1}{16}.$

$P(Y=1)=C_4^1(\frac{1}{2})^1(\frac{1}{2})^3=\frac{1}{4}.$

$P(Y=2)=C_4^2(\frac{1}{2})^2(\frac{1}{2})^2=\frac{3}{8}.$

$P(Y=3)=C_4^3(\frac{1}{2})^3(\frac{1}{2})^1=\frac{1}{4}.$

$P(Y=4)=C_4^4(\frac{1}{2})^4(\frac{1}{2})^0=\frac{1}{16}.$