please help with question 3 and 4PSTAT W120A, Summer 2016Question 4.BDue Monday Aug. 29Directions:

please help with question 3 and 4PSTAT W120A, Summer 2016Assignment 4.BDue Monday Aug. 29Directions: Please write up your solutions and submit them on GauchoSpace by the end of the dayon Monday. We prefer the submission to be single pdf file if possible. Bonus points will be awarded fororganization and neatness.Lesson 15 Cumulative Distribution Function1. The CDF of our random variable X isF (t) =t215 ? 25t + 15t2 ? 3t34for 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 Wfor values between 0 and 1 is f (w) = 1.4w, draw a graph of the CDF. Is this a valid probabilitydistribution?Lesson 16 Normal Distribution3. 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 greaterthan 500?4. For Y ? N (0.3, 0.7), calculate P{|Y ? .2| < 0.8}.Questions5. The continuous random variable R has the following probability density function on the samplespace ?1 ? r ? 1,1for ? 1 ? r ? 041f (r) = r + 4 for 0 < r ? 10elsewhereCalculate(a) P ? 12 ? R ? 21 .(b) FR (t) for t from -2 to 2.(c) E(R3 /6).6. Suppose we want to usef (y) =1,y2y ? 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 positiveand negative values of Z separately. (2) The answer must be bigger than 0.)8. (a) Find the 95th percentile of Z.1 of 2PSTAT W120A, Summer 2016Assignment 4.BDue 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 pdffX (x) = 25e?25xx ? 0.Group Question9. (a) Let Y be a discrete random variable on the natural numbers 0, 1, 2, 3, 4, . . . with a CDFFY (k). Prove that?XE(Y ) =(1 ? FY (k)) .k=0assuming that E(Y ) < ?.(b) Let X be a random variable with range [0, ?] and a continuous CDF FX (t) and a finiteexpected value. Prove thatZ ?E(X) =(1 ? FX (t)) dt.0(c) Show that for an integer r > 0,E(X r ) = rZ?tr?1 (1 ? FX (t)) dt.02 of 2