Let X be the number of students who show up for a professor's office hour on a particular day. Suppose that the pmf of X is p(0) = 0.20, p(1) = 0.25 , p(2) = 0.15, p(3) = 0.30, and p(4) = 0.10. (a) Draw the corresponding probability histogram. (Select the correct graph.) 0.30 0.30 0.25 0.30 0.30 0.25 0.20 0.25 0.20 0.25 2 0.15 0.20 2 0.15 0.20 0.10 2 0.15 0.10 2 0.15 0.05 0.10 0.05 0.10 0.00 0.05 0.0 0.05 2 3 0.00 2 3 4 0.00 3 O 2 2 3 4 X X A (b) Find the following probabilities. (i) What is the probability that at least two students show up? (ii) What is the probability that more than two students show up? (c) What is the probability that between one and three students, inclusive, show up?A chemical supply company currently has in stock 100 lb of a certain chemical, which it sells to customers in 3 lb batches. Let X equal the number of batches ordered by a randomly chosen customer, and suppose that X has the following pmf. X 1 2 3 4 p(x) 0.3 0.5 0.1 0.1 Compute E(X) . (Enter your answer in batches.) E(X) = batches Compute V(X) . (Enter your anwer in batches2.) V(X) = batches2 Compute the expected number of pounds left after the next customer's order is shipped and the variance (in lb ) of the number of pounds left. (Hint: The number of pounds left is a linear function of X.) expected weight left Ib variance of weight leftA toll bridge charges $1.00 for passenger cars and $2.50 for other vehicles. Suppose that during daytime hours, 60% of all vehicles are passenger cars. If 25 vehicles cross the bridge during a particular daytime period, what is the resulting expected toll revenue? (Enter an exact number in dollars. Hint: Let X equal the number of passenger cars; then the toll revenue h (@) is a linear function of X.) expected revenue = dollarsAn instructor who taught two sections of engineering statistics last term, the first with 25 students and the second with 40, decided to assign a term project. After all projects had been turned in, the instructor randomly ordered them before grading. Consider the first 15 graded projects. (a) What is the probability that exactly 10 of these are from the second section? (Round your answer to four decimal places.) (b) What is the probability that at least 10 of these are from the second section? (Round your answer to four decimal places.) (c) What is the probability that at least 10 of these are from the same section? (Round your answer to four decimal places.) (d) What are the mean value and standard deviation of the number of projects among these 15 that are from the second section? (Round your mean to the nearest whole number and your standard deviation to three decimal places.) mean projects standard deviation |projects (e) What are the mean value and standard deviation of the number of projects not among these first 15 that are from the second section? (Round your mean to the nearest whole number and your standard deviation to three decimal places.) mean projects standard deviation projects