Question 3 [10 marks]. Suppose that X, Y and Z are statistically independent random variables, each of them with a x²(2) distribution. (a) Find the moment generating function of U = X + 3Y + Z. State clearly and justify all steps taken. (b) Calculate the expectation E(U) using the moment generating function.

2 (Normal Distribution) Let rt be a log return. Suppose that r₁, 2, ... are IID N(0.06, 0.47). What is the distribution of rt (4) = rt + rt-1 + rt-2 + rt-3? What is P(rt (4) < 2)? What is the covariance between r2(2) = 1 + 12 and 13(2) = r² + 13? • What is the conditional distribution of r₁(3) = rt + rt-1 + rt-2 given rt-2 = 0.6?

3. Let Y be the number of speeding tickets a YSU student got last year. Suppose Y has probabilitymass function (PMF)y 0 1 2 3fY (y) 0.12 0.13 0.33 0.42(a) What is the probability a YSU student got exactly one ticket?(b) What is the probability a YSU student got at least one ticket?(c) Compute µY , the mean of Y .(d) Find the variance and standard deviation of Y .(e) What is the probability that Y exceeds its mean value?

The random variables X and Y have joint probability distribution specified by the following table: y=0 y=1 y=2 1 1 x=0 4 8 1 3 x=1 4 8 Please provide all answers to the following to three decimal places. (a) Find the expectation of XY. 0.375 (b) Find the covariance Cov(X,Y) between X and Y. 1 64 (c) What is the correlation between X and Y? -0.0463 (d) Suppose the random variables X and Y above are connected to random variables U and V by the relations X = 9U + 9 Y = 4V + 8 || What is the covariance Cov(U,V)? 1 5184 (e) What is the correlation between U and V?

Suppose that X,Y and Z are stat istically independent Question random variables, each of them with a x²(2) distribution. (a) Find the moment generating function of U = X + 3Y + Z. State clearly and justify all steps taken. %3D (b) Calculate the expectation E(U) using the moment generating function. Hint: You may use without proof the fact that the moment generating function of a x*(v) random variable W is Mw(1) = (G==)". 2t

qmplus.qmul.ac.uk Question 3 Suppose that X, Y and Z are statistically independent random variables, each of them with a x²(2) distribution. (a) Find the moment generating function of U = X +3Y + Z. State clearly and justify all steps taken. (b) Calculate the expectation E(U) using the moment generating function. Hint: You may use without proof the fact that the moment generating function of a x²(v) random variable W is Mw (t) = (1-21) 2t v/2 Continue to next page A

gn X Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is 0.41 (a couple will travel together in the same vehicle, so either both people will be on time or else both will arrive late). Assume that different couples and individuals are on time or late independently of one another. Let X = the number of people who arrive late for the seminar. (a) Determine the probability mass function of X. [Hint: label the three couples #1, #2, and #3 and the two individuals #4 and #5.] (Round your answers to four decimal places.) P(X = X) X 0 1 2 3 4 5 6 7 8 (b) Obtain the cumulative distribution function of X. (Round your answers to four decimal places.) F(x) X 0 1 2 3 4 5 6 7 Use the cumulative distribution function of X to calculate P(2 ≤ x ≤ 5). (Round your answer to four decimal places.) P(2 ≤ x ≤ 5) = Need Help? Read It Watch It 4

Q5. Please see attached.

(c) Find the expected value of Y. (d) Find the variance of Y.

K In finance, one example of a derivative is a financial asset whose value is determined (derived) from a bundle of various assets, such as mortgages. Suppose a randomly selected mortgage in a certain bundle has a probability of 0.07 of default. (a) What is the probability that a randomly selected mortgage will not default? (b) What is the probability that nine randomly selected mortgages will not default assuming the likelihood any one mortgage being paid off is independent of the others? Note: A derivative might be an investment that only pays when all nine mortgages do not default. (c) What is the probability that the derivative from part (b) becomes worthless? That is, at least one of the mortgages defaults. (a) The probability is 0.93 (Type an integer or a decimal. Do not round.) (b) The probability is 0.5204 (Round to four decimal places as needed.) (c) The probability is (Round to four decimal places as needed.) .…