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 P(x, y) 1 2 0.010 0.015 0.025 1 0,020 0.030 0.050 0.050 0.075 0.125 3 0.060 0.090 0.150 4. 0.040 0.060 0.100 0.020 0.030 0.050 (a) What is the probability that there is exactly one car and exactly one bus during a cycle? (b) What is the probability that there is at most one car and at most one bus during a cycle? (c) What is the probability that there is exactly one car during a cycle? Exactly one bus? P(exactly one car) = P(exactly one bus) = (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? (e) Are X and Y independent rv's? Explain. O Yes, because p(x, y) = Px(x) · P{Y). O Yes, because p(x, y) * P(x) Pylv). O No, because p(x, y) = P(x) Py(Y). O No, because p(x, y) + Py(x) Py(Y). Need Help? Read It Watch It

2 3. Consider the joint probability distribution of X and Y: x = -2 0.25 x = 0 x = 2 y = -4 y = 0 0.25 0.5 In this example, X and Y are not independent, correlated not independent, uncorrelated independent, correlated independent, uncorrelated In this example, X and Y are 4. Consider the joint probability distribution of X and Y: x = 0 x = 1 y = 0 0.2 0.2 y= 1 0.1 0.5 In this example, X and Y are not independent, correlated not independent, uncorrelated independent, correlated independent, uncorrelated In this example, X and Y are

Data were collected from a survey given to graduating college seniors on the number of times they had changed majors. From that data, a probability distribution was constructed. The random variable X is defined as the number of times a graduating senior changed majors. It is shown below: xx  0 1 2 3 4 5 6 7 8 P(X=x)P(X=x)  0.244 0.234 0.233 0.16 0.068 0.039 0.017 0.003 0.002 What is the probability that a randomly selected student changed his or her major at least once?  What is the probability that a randomly selected student changed his or her major at most twice?  What is the probability that a randomly selected student changed his or her major exactly twice?  What is the mean number of times students changed majors?   What is the standard deviation of X?

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. p(x, у) 1 2 0.015 0.010 0.025 1 0.030 0.020 0.050 2 0.075 0.050 0.125 3 0.090 0.060 0.150 4 0.060 0.040 0.100 0.030 0.020 0.050 (a) What is the probability that there exactly one car and exactly one bus during a cycle? (b) What is the probability that there is at most one car and at most one bus during a cycle? (c) What is the probability that there is exactly one car during a cycle? Exactly one bus? P(exactly one car) P(exactly one bus) = (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? (e) Are X and Y independent rv's? Explain. Yes, because p(x, y) = Px(x) · Py(y). Yes, because p(x, y) ± Px(x) · py(y). No, because p(x, y) = Px(x) · Py(Y). No, because p(x, y) # Px(x) · Py(y).