1.1 Suppose that the random variable x follows a lognormal distribution such that its 90th percentile is 95.88059 and its 99th percentile is 774.87305. Determine its 95th percentile. 1.2 The cumulative distribution function of the Weibull distribution is F(x;k,λ)-1-exp (11) Derive f(x), the quantile function, expected value, variance, skewness and moment generating function (m.g.f.). (14)

USA Today reported that about 20% of all people In the United States are Illiterate. Suppose you take eleven people at random off a city street. P USE SALT (a) Make a histogram showing the probability distribution of the number of illiterate people out of the eleven people in the sample. 0.30 0.30 0.25 0.25 0.20 0.20 0.15 0.15 0.10 0.10 0.05 0.05 2 3 4 5 7. 8 9 10 11 0 1 2 3 4 5 6 78 9 10 11 0.30 0.30 0.25 0.25 0.20 0.20 0.15 0.15 0.10 0.10 0.05 0.05 0. 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 45 6 7 8 9 10 11 (b) Find the mean and standard deviation of this probability distribution. (Round your standard deviation to two decimal places.) people people Find the expected number of people in this sample who are illiterate. people

The random variable X takes value O with probability, and value 1 with the remaining probability. The random variable Y takes value 100 with probability and value 101 with the remaining probability. What is the relation between the variance of X and the variance of Y? Var[X] Var[Y] "

A,b,c,d.

Let X be a normal random variable with mean 85 and a variance of 25 (i.e., X ∼ N (85, 25)). step1:  Let M = aX + b for some constants a, b not equal to 0. Write an expression for the pdf of M . step 2:  Suppose that X represents an approximate distribution of the final scores in a certain math course (ignore the fact that this approximation can technically have scores that are greater than 100 or less than 0). If the teacher were to curve the scores, it means he would determine a function to apply to the scores to achieve a desired distribution (assume an affine function in this case, as in the previous part). Suppose he wants the scores to be normally distributed with mean 80 and a standard deviation of 4. What should he choose for a and b? What students would see their score lowered, and what students would see their score increased?

3 The median value y of a continuous random variable is that value such that F(y) = 0.5. Find the median value of the random variable in 90.5 = f(y) = 2 25 -Y, 0 ≤ y ≤ 5, elsewhere. 0,

Suppose X, Y, and Z are three independent normal random variables. X has an expected value of -7 and a standard deviation of 1.6; Y has an expected value of -9 and a standard deviation of 1.8; Z has an expected value of 8 and a standard deviation of 1.4 Let T = 2X + 4Y+7Z Variable Mean(Expected Value) Standard Deviation Coefficient X Y -7 -9 1.8 4 1.6 2 Z 8 1.4 7

Consider the distribution of a discrete variable Y whose cumulative distribution function is given by 0, y < 12 12 ≤ y < 32 32 ≤ y ≤ 48 1, 48 ≤ y Enter below the variance of this distribution. F(y) 0.151, 0.699,

A stock price is currently $40. It is known that at the end of 1 year it will be either $48or $32. The risk-free Interest rate is 5% per annum with continuous compounding. What is the value of a portfolio today with a strike price of $527 (use A-0.25) Hint: draw a tree