Question 1 There are two traffic lights on the way to work L

Question 1: There are two traffic lights on the way to work. Let X1 be the number of lights that are red, requiring a stop, and suppose that the distribution of X1 is as follows: x 0 1 2 P(Xx) 0.2 0.5 0.3 Let X2 be the number of lights that are red on the way home; X2 is independent of X1. Assume that X2 has the same distribution as X1, so that X1, X2 is a random sample of size n = 2. Let T = X1 + X2, and determine the sampling distribution of T.

Solution

P(T = 0) = P(X1 = 0)*P(X2 = 0) = 0.2*0.2 = 0.04

P(T=1) = P(X1 = 0)*P(X2 = 1) + P(X1=1)*P(X2 = 0) = 0.2

P(T=2) = P(X1=2)*P(X2=0) + P(X1 = 1)*P(X2=1) + P(X1=0)*P(X2=2) = 0.06+0.25+0.06 = 0.37

P(T=3) = P(X1=2)*P(X2=1) + P(X1=1)*P(X2=2) = 0.3

P(T=4) = P(X1=2)*P(X2=2) = 0.09