The following model is proposed for the distribution of family size in a large population:
P(k children in family;$\theta$) = $\theta^{k}$, for $k = 1, 2, ...$
P(0 children in family;$\theta$) = $\frac{1-2\theta}{1-\theta}$.
I tried to multiply them together and got the likelihood function
$L(\theta)=\frac{1-2\theta}{1-\theta}\prod_{k=1}^\infty \theta^{k}$
I don't think it is the right way to approach. Any help is appreciated.
Likelihood function is a function of samples $x_1,\ldots,x_n$ and of $\theta$.
$$ f(x_1,\ldots,x_n|\theta)=f(x_1|\theta)\times\dots\times f(x_n|\theta). $$ Here p.m.f. $f(x_i|\theta)$ is equal to $$ f(x_i|\theta) = \left(\frac{1-2\theta}{1-\theta}\right)^{1_{\{x_i=0\}}}\times \theta^{x_i}, $$ where $1_{\{x_i=0\}}=1$ if ${x_i=0}$ and zero otherwise.
Next, multiplying p.m.f. for different samples get $$ f(x_1,\ldots,x_n|\theta)= \prod_{i=1}^n\left(\frac{1-2\theta}{1-\theta}\right)^{1_{\{x_i=0\}}}\times \theta^{x_i}=\left(\frac{1-2\theta}{1-\theta}\right)^{S_0}\times \theta^{\sum_{i=1}^n x_i}, $$ where $S_0=\sum_{i=1}^n 1_{\{x_i=0\}}$ is the number of zeros in the samples.
Next, calculate logarithm and MLE.