Define MLEs for the following problems: (a) Estimating the p i s p i s based on a random sample from a multinomial

Define MLEs for the following problems:

(a) Estimating the $p_i^{\prime }s$ based on a random sample from a multinomial population distribution.

(b) Estimating the unknown parameters in the mean vector $\mu$ and covariance matrix $\Sigma$ based on a random sample from a bivariate normal distribution.

(c) Estimating $\beta$ in the population distribution

$f(x;\beta )=\frac{2x}{\beta }\exp \left[\frac{-x^2}{\beta }\right]I_{(0,\infty )}\left(x\right)$

(d) Estimating the parameter $p$ in a negative binomial population distribution where $r$ is known.