Problem 2 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 = 5. (Recall that if X~ Poisson (A), then P(X = n)= " for each n ≥ 0.) Assume that the number of the 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. What is the distribution of the number of games she plays to get her first (b) medal? 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?

Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)

25. Let X have a three-dimensional normal distribution with mean vector and covariance matrix and A- 3-1 respectively. Set Y= X1 +X + X, and Y= X1+X. Determine the conditional distribution of Y given that Y 0.

1. The cumulative probability function of a discrete random variable X is given in the following table: 1 2 3 f(x) 1/8 3/8 3/8 1/8 a. Find the distribution function F(x) for the random variable X. b. Graph this distribution function.

Part IV: Sections 5.5-5.7 17. Suppose that the number of defects in a randomly selected one carat diamond of a certain type follows a Poisson distribution with X= 3. By the Central Limit Theorem, what is the approximate (a) distribution of the mean number of defects in the random sample of 40 one carat diamonds? You should give a name for the distribution, as well two associated numerical quantities. (b) the random sample of 40 one carat diamonds, the probability that the sample mean number of defects is less than y is less than 98%. Estimate the largest number of defects y such that in