Finding likelihood function and maximum likelihood estimate

Question

Genetic theory shows that in a population in which the gene frequency for colour blindness is $\theta$ then the probability that a male is colour blind is θ and the probability that a female is colour blind is $\theta^2$ . A random sample of 50 males is found to include 5 who are colour blind.Write down the likelihood function and obtain the maximum likelihood estimate $\hat{\theta}_m$. A second independent random sample of 100 females is found to include 4 who are colour blind. Again write down the likelihood function and obtain the maximum likelihood estimate $\hat{\theta}_f$ . Now find the likelihood function for $\theta$ based on both samples and obtain the combined maximum likelihood estimate $\hat{\theta}$. Comment on what you see.

Answer 1

Hint: Given that the probability of being a color blind man is $\theta$, the likelihood of a sample with $n_{m}$ men out of which $b_{m}$ are color blind is $\theta^{b_{m}}(1-\theta)^{n_{m}-b_{m}}$. By maximizing logarithm of this expression with respect to $\theta$, one gets $\hat{\theta}^{MLE}(n_{m},b_{m})=\frac{b_{m}}{n_{m}}$. Using similar approach for women and the two samples combined, you can get the other estimates. I believe the three estimates should be $\frac{1}{10}$, $\frac{1}{5}$ and $\frac{1}{100}(\sqrt{601}-9)$.