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 $X$?