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Problem 1. Suppose a random sample

`x_(1),dots,x_(n)?Bernoulli(p)`

with

`0<=p<=1`

. Find the maximum likelihood estimator of

`p`

. (a) Find the

`log`

likelihood function for

```
p=0,1hat(p)=\sum_(i=1)^n (x_(i))/(n)l(p)p=0hat(p)=\sum_(i=1)^n (x_(i))/(n)0.
(d) We can find that the likelihood is defined at p=0 or 1 . Show that hat(p)=\sum_(i=1)^n (x_(i))/(n) is also the
maximizer of the likelihood.0.
(c) Use Theorem 5.2 to conclude that hat(p)=\sum_(i=1)^n (x_(i))/(n) is the maximizer of l(p) for 0.
(d) We can find that the likelihood is defined at p=0 or 1 . Show that hat(p)=\sum_(i=1)^n (x_(i))/(n) is also the
maximizer of the likelihood.0. Note that if p=0,1, the log likelihood is not
defined.
(b) Calculate the first and the second derivatives of the log likelihood function for 0.
(c) Use Theorem 5.2 to conclude that hat(p)=\sum_(i=1)^n (x_(i))/(n) is the maximizer of l(p) for 0.
(d) We can find that the likelihood is defined at p=0 or 1 . Show that hat(p)=\sum_(i=1)^n (x_(i))/(n) is also the
maximizer of the likelihood.
```