The joint probability distribution of two random variables X, Y is given in the table below. Y=-4 Y=-2 Y=0 Y=2 Y = 4 X=0 0.02 0.07 0.11 0.03 0.08 X=2 0.00 0.11 0.09 0.12 0.00 X=3 0.19 0.05 0.00 0.03 0.10 From the information in the table, calculate each of the following three probabilities. (a) P(X<2, Y≥-2)-0 (b) P(x < 0) = (c) P(X<3)= X Ś

Fill in the  P=Xx  values to give a legitimate probability distribution for the discrete random variable  X , whose possible values are  −1 ,  3 ,  4 ,  5 , and  6 . ValuexofX P=Xx −1   3 0.10 4   5 0.21 6 0.10

Fill in the  P=Xx  values to give a legitimate probability distribution for the discrete random variable  X , whose possible values are  1 ,  3 ,  4 ,  5 , and  6 . ValuexofX P=Xx 1 0.15 3 0.29 4 0.21 5   6

The joint probability distribution of the number X of cars and the number Y of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table. y 1 p(x, y) 0 1 0.030 0.020 50 X 3 0.090 0.060 4 0.060 0.040 2 0 0.015 0.010 0.025 0.050 2 0.075 0.050 0.125 0.150 0.100 5 0.030 0.020 0.050 (a) What is the probability that there is exactly one car and exactly one bus during a cycle? 0.020 (b) What is the probability that there is at most one car and at most one bus during a cycle? 0.075 (c) What is the probability that there is exactly one car during a cycle? Exactly one bus? P(exactly one car) = 0.220 x P(exactly one bus) = 0.200 (d) Suppose the left-turn lane is to have a capacity of five cars and one bus is equivalent to three cars. What is the probability of an overflow during a cycle? 0.985 X (e) Are X and Y independent rv's? Explain. O Yes, because p(x, y) = P(x) Py(Y). O Yes, because p(x, y) = Px(x) • P₂(Y). # ● No, because p(x, y) = Px(x) •…