Calculate the maximum likelihood estimator, $\tilde{p}_{ML}$, of $p$

Question

This problem was part of an assignment in my undergrad stats class.
Suppose Steph Curry takes ten 3-point shots per game, and the number of 3-point shots scored follows a $𝐵𝐼𝑁\sim(10, 𝑝)$ distribution, where 𝑝 is the probability of successfully scoring a 3-point shot. [Note: in basketball a 3-point shot is one taken from behind the 3- point arc]. To estimate this probability, a random sample of 10 games is considered and the number of successful 3-point shots is recorded. These data are shown below.

$$\begin{array}{ccccc} \hline 5 & 3 & 0 & 7 & 10 \\ 2 & 4 & 6 & 8 & 1 \\ \hline \end{array}$$

Calculate the maximum likelihood estimate, $\tilde{p}_{ML}$, of $p$, the probability of scoring a 3- point shot, and verify (using either the first or second derivative test) that you have found a maximum.

So I've $X\sim BIN(10,p)$

I go through the steps to find the estimator and arrive at

\begin{equation} \sum_{i=1}^{10}x_i(1-p)=p\sum_{i=1}^{10}(10-x_i) \end{equation}

I'm at a standstill in how I solve for $p$. Then when I have the estimator, do I do anything with the provided info in the table? any help would be appreciated

Answer 1

1. Start by bringing the $1-p$ term outside the sum, $$ (1-p)\sum_{i=1}^{10} x_i = p\sum_{i=1}^{10}(10-x_i)$$

Upon using the distributive property on $(1-p)\sum_{i=1}^{10}x_i$, you will see that both sides of the equation have a $-p\sum_{i=1}^{10}x_i$ that can cancel out. You should be able to solve for $p$ from there. You may also need to know that $\sum_{i=1}^{a}b = a*b$, where $a$ is a fixed positive integer and $b$ is a fixed arbitrary real number.

2. Don't forget to apply the second derivative test to show that the critical point is a maximum.

3.To compute the MLE estimate: When you solve for $p$, you will get a formula for $\tilde{p}_{ML}$. The numbers in the table correspond to the values of $x_i$ in that table. Plug them in.

Hope this helps.