STAT 67, Fall 2016 Homework 4 11-10-2016 1. You are waiting for the bus to arrive to pick you up. You believe the wait time can be any number between 0 and 5 minutes, with uniform density. a. What is the probability that you wait less than 2.5 minutes? b. What is the expected time in minutes that you will be waiting for the bus? Interpret this in context of the problem in a sentence or two. c. What is the probability that you wait between 1.5 and 3.5 minutes? d. Now say you wait for the bus 5 days in a row, with each day being independent of one another. What is the probability that you waited more than 2.5 minutes everyday? e. Continuing the scenario in part d., what is the probability that you waited more than 2.5 minutes at least one of the days? 2. Cars arrive at an inspection station at a rate of 10 cars per hour. With probability of 0.5 a car will pass inspection. a. What is the expected number of cars to arrive at the inspection station in 1 hour? What is the expected number of cars to arrive in a 3 hour period? b. What is the probability that at most (less than or equal to) 15 cars will arrive in the 1 hour time period? What is the probability that at most 45 cars will arrive in the 3 hour time period? c. In a 3 hour time period, what is the probability that more than 4 cars will arrive? d. Say the cars are independent of one another. What is the probability that 10 cars will arrive in a 1 hour time period AND all 10 will pass inspection. Hint: P(A and B)=P(A)*P(B|A) 3. Let f (x) = cx3 and SX = [0, 2] (X is continuous). a. What value of c will make f (x) a valid density? b. What is P (X = 1)? What about P (X = 1 or X = 2)? i c. Find E(X). d. What is P (0.5 < X < 1.5)? 4. You are to wait for the first car to arrive at the station. It is believed that the wait time follows an Exponential distribution with parameter = 10 minutes. a. What is the probability that you wait for more than 3 minutes? b. What is the expected time (in minutes) that you wait for the first car to arrive? c. Now say you have already waited for 2 minutes. What is the probability that you will have to wait at least an additional 3 minutes for a car to arrive? d. You are to wait for 10 cars to arrive. How long do you expect to wait to complete this task? How much do you expect your wait time to vary around the expected time? 5. You are to model the daytime temperature of the Mojave desert during the summer time. Temperatures in the desert are modeled to follow a normal distribution with = 102 Fahrenheit and variance 2 = 25 Fahrenheit. a. Write out the integral form of the probability that the temperature will be between 97 to 107 degrees Fahrenheit. Use pnorm function in R, pnorm(x=120, , ) or use empirical rule to compute numeric value. b. What is the probability that the temperature will be 120 degrees? What is the probability of the temperature being below 120? Write out integral form and use R (pnorm) to compute. c. Say X is the random variable that denotes the temperature during the desert day. What distribution does 0.55X-17.6 follow? State the type of distribution, and all parameters. d. Now say you relax the normality assumption from the model, and you observe 25 days (X1 , X2 , ..., X25 or to say Xi for i=1,2,3,...,25). What is the approximate distribution of the sample n = 1 P Xi ? State the approximate type of distribution and all mean of these daily temperatures, X n parameters. i=1 e. You now transform each temperature from Fahrenheit to Celsius. The conversion is Fahrenheit = 0.55Fahrenheit-17.6. What approximate distribution does the sample mean in Celsius follow? State the type and parameters. Hint: The sample mean in celsius is the average of the transformed variables, Y = 17.6)). ii n 1P (0.55Xi n i=1