Suppose a random sample of size n is drawn from probability model $$p_X(x;\theta) = \frac{\theta^{2x}e^{-\theta^2}}{x!}\;, \quad x=0, 1, 2, ...$$ Find a forumla for the maximum likelihood estimator, $\hat{\theta}$
$$L(\theta) = \prod_{i=1}^{n}\frac{\theta^{2x_i}e^{-\theta^2}}{x_i!}$$
$$L(\theta) = \frac{\theta^{2\sum_{i=1}^{n}x_i}e^{-n\theta^2}}{x_1!x_2!...x_n!} $$
$$ln[L(\theta)] = 2\sum_{i=1}^{n}x_iln(\theta)-n\theta^2 - ln(x_1!x_2!...x_n!)$$
$$T = \frac{d\big[ln[L(\theta)]\big]}{d\theta} = \frac{2\sum_{i=1}^{n}x_i}{\theta}-2n\theta$$
Let $T=0$
$$2n\theta = \frac{2n\bar{x}}{\theta}$$
$$\theta = \sqrt{\bar{x}}$$
This implies that $\hat{\theta} = \sqrt{\bar{x}}$
I have not got an answer in the back of the book for this one, does anybody see anything wrong with it?
$$T' = \frac{-2n\bar{x}}{\theta^2} - 2n$$
Which is negative for all $\theta$ because $x \ge 0$, thus the likelihood is a maximum.
Your answer is correct. You can also check the second derivative to confirm maximality.