Maximum Likelihood Estimates of θ

Question

Q: Players kill monsters for items. When a monster is killed, it drops an item which can be one of three types, with the following probabilities:

P(Legendary item) = $\theta_1$

P(Rare item) = $(1 − \theta_1)\theta_2$

P(Magical item) = $(1 − \theta_1)(1 − \theta_2)$

Suppose a player kills n monsters. Let $X = (X_1, X_2, X_3)$ be the number of legendary, rare, and magical items respectively. Assuming the drops are independent, What is the probability mass function of X and what are the MLE's of $\theta_1$ and $\theta_2$?

Does $X$ has binomial distribution? can i find the MLE by putting summation in front of the density function of $X$?

Answer 1

$X$ follows multinomial distribution.

$$P(X=(n_1,n_2,n_3)) = \frac{n!}{n_1!n_2!n_3!}\theta_1^{n_1}((1-\theta_1)\theta_2)^{n_2}((1-\theta_1)(1-\theta_2))^{n_3}1_{n_1+n_2+n_3=n}$$

To compute MLE, assuming the values of $n_1,n_2,n_3$ are known, follow the procedure in this answer, to get,

$$\theta_1 = \frac{n_1}{n}, \ \theta_2 = \frac{n_2}{n-n_1}$$