A financial company recommends the subsequent method for the game of roulette. It recommends that the gambler wager 1 on purple. If pink seems (which has opportunity 18/38 of occurring) then the gam- bler have to take his income of 1 and stop. If the gambler loses this bet, he must then make a 2d bet of size 2 and then cease. Let X denote the gambler's winnings. (a) Find PX > zero. (b) Find E[X]. Exercise 1.9 Four buses wearing 152 college students from the identical school arrive at a soccer stadium. The buses carry (respectively) 39, 33, 46, and 34 students. One of the 152 college students is randomly chosen. Let X denote the variety of college students who had been on the bus of the selected stu- dent. One of the 4 bus drivers is also randomly chosen. Let Y be the variety of college students who have been on that driving force's bus. (a) Which do you believe you studied is larger, E[X] or E[Y ]? (b) Find E[X] and E[Y ]. Exercise 1.10 Two gamers play a tennis match, which ends while one of the players has won two sets. Suppose that each set is similarly probable to be gained through both participant, and that the results from distinct sets are independent. Find (a) the anticipated price and (b) the variance of the quantity of sets performed.

Suppose that - in any given term - a positive inventory is equally in all likelihood to move up 1 unit or down 1 unit, and that the consequences of various durations are impartial. Let X be the quantity the stock is going up (either 1 or ?1) in the first period, and let Y be the cumulative quantity it is going up within the first three intervals. Find the correlation among X and Y. Exercise 1.19 Can you assemble a pair of random variables such that Var(X ) = Var(Y ) = 1 and Cov(X, Y ) = 2? Exercise 1.20 If Y is a random variable and h a feature, then h(Y ) is likewise a random variable. If the set of wonderful possible values of h(Y ) are hi,i ? 1, then by means of the definition of predicted cost, we have that E[h(Y )] = i hello P(h(Y ) = hi). On the other hand, because h(Y ) is same to h(y) while Y = y, it is intuitive that E[h(Y )] = -

y h(y)P(Y = y) Verify that the previous equation is legitimate.

QUESTION 1

A standard normal density is a probability density function for a random variable that is normally distributed with population mean = 0 and population variance = 1. Using a table of standard normal probabilities, such as Table A1 in your textbook, answer the following questions: a) If a random variable has a standard normal density, what is the probability that this random variable is less than 1.5? b) If a random variable has a standard normal density, what is the probability that this random variable is less than -2.0? c) If a random variable has a standard normal density, what is the probability that this random variable is between -2.0 and 1.5? d) Find the 68% critical value for a random variable that has a standard normal density.