PSTAT W120A, Summer 2016 Assignment 4.B Due Monday Aug. 29 Directions: Please write up your solutions and submit them on GauchoSpace by the end of the day on Monday. We prefer the submission to be single pdf file if possible. Bonus points will be awarded for organization and neatness. Lesson 15 Cumulative Distribution Function 1. The CDF of our random variable X is F (t) = \u0001 t2 15 25t + 15t2 3t3 4 for 0 t 2. (a) Calculate the probability of { 21 X 1}. (b) Find the density of X. 2. For a random variable W where P{W = 0} = 0.1 and P{W = 1} = 0.2 and the density of W for values between 0 and 1 is f (w) = 1.4w, draw a graph of the CDF. Is this a valid probability distribution? Lesson 16 Normal Distribution 3. A SAT score is designed to have a normal distribution with mean 400 and standard deviation 200. If we take 5 independent SAT scores, what is the probability that the mean of them is greater than 500? 4. For Y N (0.3, 0.7), calculate P{|Y .2| < 0.8}. Questions 5. The continuous random variable R has the following probability density function on the sample space 1 r 1, 1 for 1 r 0 4 1 f (r) = r + 4 for 0 < r 1 0 elsewhere Calculate \b (a) P 12 R 21 . (b) FR (t) for t from -2 to 2. (c) E(R3 /6). 6. Suppose we want to use f (y) = 1 , y2 y 1. (a) Show that this is a valid density function for a random variable Y . (b) Show that EY does not exist. 7. Calculate E|Z| where Z is the standard normal. (Hints: (1) Calculate the integrals for positive and negative values of Z separately. (2) The answer must be bigger than 0.) 8. (a) Find the 95th percentile of Z. 1 of 2 PSTAT W120A, Summer 2016 Assignment 4.B Due Monday Aug. 29 (b) Find the 95th percentile of a normal distribution with mean 100 and standard deviation 20. (c) Find the 25th percentile of a normal distribution with mean 10.5 and variance 65. (d) Find the 5th percentile of the exponential random variable X which has pdf fX (x) = 25e25x x 0. Group Question 9. (a) Let Y be a discrete random variable on the natural numbers 0, 1, 2, 3, 4, . . . with a CDF FY (k). Prove that X E(Y ) = (1 FY (k)) . k=0 assuming that E(Y ) < . (b) Let X be a random variable with range [0, ] and a continuous CDF FX (t) and a finite expected value. Prove that Z E(X) = (1 FX (t)) dt. 0 (c) Show that for an integer r > 0, E(X r ) = r Z tr1 (1 FX (t)) dt. 0 2 of 2