Question

Suppose it is known from large amounts of historical data that X, the number of cars that arrive at a specific intersection during a 20-second time period, is characterized by the following discrete probability function: 6* f(x) = e-6 for x = 0, 1, 2,.. a. Find the probability that in a specific 20-second time period, more than 8 cars arrive at the intersection. b. Find the probability that only 2 cars arrive. c. Find the expectation E[X] and Var(X) of the random variable X.

Answer 1

e-6 6X XI X : the number of cars that arrive at specific intersection during a go-second time period. Then, X ~ Poisson (6) P(x=x) f(x) 9 X= 0, 1, 2,000 (a) We want : P(X)8) 1 - P(X37) -1- PIX=D) – P(X=1) - PLX = 2) - P(X=3) P(x = 4) – PlX= 5) - P(x=6)- P(X=7) 677 1- et both 11 21 7! =trebx 300.1428571 - 0.74398 P(X78) 0.25602 62 + + 2 (b) PCX = 2) e-6 0,0446175 2! (6) since , This is a poisson distribution o E(X) 6 Var(x) 6 so,