Suppose that the number of children N of a randomly chosen family satisfies
$P(N=n) = \dfrac{3}{5} \left(\dfrac{2}{5}\right)^n $
for $n=1,2,3 ...$
Now suppose that a child is equally likely to be a girl or a boy, and let X be the number of daughters in a randomly chosen family.
I am trying to find $E[X]$ (By the way I found $E[N]=\frac{2}{3}$)
My initial thought is to use $E[X|N = n]$ Because, the number of daughters in family of $n$ children is the random variable $X$ and it is binomial distribution with parameter $n,p$ and as we know expected value in this case $\frac{1}{2}$ . Therefore our $E[X|N=n]=np=\frac{1}{2}$n
Now I am stuck here. I am thinking to use $E[E[X|N=n]]$ but I am not sure if this is appropriate for calculation $E[X]$.
That is exactly correct. Because the conditional variable $X \mid N = n$ is binomial with parameters $n$ and $p = 1/2$, you computed $$\operatorname{E}[X \mid N] = N/2,$$ hence $$\operatorname{E}[X] = \operatorname{E}[\operatorname{E}[X \mid N]] = \operatorname{E}[N/2] = \operatorname{E}[N]/2.$$ What are your reservations regarding this approach?