1
$\begingroup$

Jimmy Butler (shooting guard for the Minnesota Twolves) makes about p =30% of his three-point shot attempts. In any game, the number of three-point shots he attempts is a random variable Z whose distribution is approximately Poisson with mean = 5. What is the probability that in a random game he makes exactly 2 three-point shots?

I can easily find the probability of taking n shots given the distribution, but I have trouble finding how to find the probability of making n shots, since p = .3, and 6.6666 attempted shots is not a valid input for the Poisson distribution formula.

Any help is appreciated!

$\endgroup$
2
$\begingroup$

This is a distribution mixture problem. Given the number of shots $n$ he attempts, the number of shots he makes $m\sim B(n,p)$ is governed by a binomial distribution with $p=0.3$. Then the number of shots $n\sim\mbox{Poisson}(\lambda)$ he attempts is governed by a Poisson distribution with $\lambda=5$. The goal is to find the mixed distribution for $m$, and in particular, the probability of $P(m=2)$. Give it a try!

Hint: show that $\,m\sim\mbox{Poisson}(p\lambda)\,$ after the mixture.

$\endgroup$
1
$\begingroup$

Let $N$ be the number of shots he takes and $M$ be the number of shots he makes.

The thing you find it easy to compute is $P(N=n).$

You should also be able to write down the probability he makes $m$ shots given that he takes $n$ since you know that the $n$ shots are independent trials with probability $p=0.3.$ This is the conditional distribution of $M$ given $N$, denoted as $P(M=m\mid N=n).$

You want $P(M=m),$ the distribution of the number of shots he makes, which can be obtained from the above quantities via the law of total probability as $$ P(M=m) = \sum_{n=0}^\infty P(M=m\mid N=n)P(N=n).$$

$\endgroup$
  • $\begingroup$ Thanks! That was very useful. $\endgroup$ – user493197 Oct 19 '17 at 17:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.