Question

need to check my work. Just need B and C

Problem 2. Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is fx (x) = e-λ- XE(0, 1,2, ) ar! This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a Prove by direct cornputation that the mean of a Poisson randoln variable with parameter λ is simply b) Let Xi ~Poisson(Xi) and X2 ~ Poisson(A2) be independent. Prove that the random variable X1+X2 is also Poisson with parameter λ1 + λ2 Hint 1: The Persian mathematician Al-Karaji was the first to write down what is today known as the binomial formula: if n E 10,1,2,...), then rL rL n- You will likely find the binomial formula useful in your proof Hint 2: For Xit X2 to be x, we can have X, = 0 and X2 2, or X1-1 and X2 = x-1, or X1 and X2-a- 2, and so on. How do we compute P(X1 + X2-r)? 2 c) Suppose that the average number of accidents on the I-5 per day is 5, and that the average number on the I-8 is 11. Assume that the number of accidents on each are independent Poisson random variables What is the probability that the total number of accidents in a day between both expressways is less than or equal to 9? You may use a computer to calculate a decimal answer. For example, scipy.stats.poisson might be useful. But if you use a computer, give the code that you used

Answer 1

If メ、 tam X~ Poisson メ." Poisson(M.) Mdepnd 2