## Problem 2Alice is a goaltender in an ice hockey team. Suppose that the number of goals she concedes each game is distributed as a Poisson distribution with parameter λ = 5. (Recall that if \( X \sim \text{Poisson}(\lambda) \), then \( P(X = n) = \frac{e^{-\lambda} \lambda^n}{n!} \) for each \( n \geq 0 \).) Assume that the number of goals she concedes at different games are independent of each other.Alice gets a medal after each game if and only if she concedes zero or one goal in that game.(a) **What is the distribution of the number of games she plays to get her first medal?**(b) **What is the expected number of games she plays to get her third medal?**(c) **What is the probability that she will receive strictly more than two medals in the first 30 games?**

Given that Alice is a goaltender in an ice hockey team. Suppose that the number of goals she concedes each game is distributed as a Poisson distribution with parameter .(a) Let the random variable X denote the number of goals she concedes each game. Then The pmf of X isAssume that the number of goals she concedes at different games are independent of each other. Alice gets a medal after each game if and only if she concedes zero or one goal in that game.The distribution of the number of games she plays to get her first medal is Geometric distribution with the probability of getting a medal Let Y denote the number of games she plays to get her first medal. Then, .The PMF of Y is(b) The number of games Alice plays until she gets her third medal follows a negative binomial distribution.Let the random variable Z denote the number of games Alice plays until she gets her rth medal out of total n games.The pmf of Z is where and The expected number of games she plays to get her rth medalTherefore, the expected number of games she plays to get her third medal is 74.2(c) The random variable T denotes the number of medals in 30 games. The T follows Binomial distribution with and .The PMF of T isThe probability that she will receive strictly more than two medals in the first 30 gamesTherefore, the probability that she will receive strictly more than two medals in the first 30 games is 0.1197.(a) The distribution of the number of games she plays to get her first medal is Geometric distribution with the probability of getting a medal .(b) The expected number of games she plays to get her third medal is 74.2(c) The probability that she will receive strictly more than two medals in the first 30 games is 0.1197.