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.

Let X be a random variable with pdf f(x) = 0.2 €-0.2x 0 if x > 0, otherwise. (a) Find the moment generating function of X, remembering to state the range for which it holds. (b) Hence find the expectation and variance of X.

The discrete random variables X and Y have the following joint probability function Y 0 1 2 1 1/20 2/20 3/20 X 2 2/20 1/20 4/20 3 3/20 1/20 3/20

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10. Let and be independent random variables representing the lifetime (in 100 hours) of Type A and Type B light bulbs, respectively. Both variables have exponential distributions, and the mean of X is 2 and the mean of Y is 3. a) Find the joint pdf f(x, y) of X and Y. b) Find the conditional pdf f₂ (ylx) of Y.. c) Find the probability that a Type A bulb lasts at least 300 hours and a Type B bulb lasts at least 400 hours. d) Given that a Type B bulb fails at 300 hours, find the probability that a Type A bulb lasts longer than 300 hours. e) What is the expected total lifetime of two Type A bulbs and one Type B bulb?

Mx(t)=ebt-eat/t(b-a) Calculate the expected value and variance of the random variable X using the moment generating function.

If the moment generating function of a random variable X is: (1/3+(2/3)e t ) 5 find P (X > 3).

Let X be a Geometric random variable with parameter p = 1/4. Find E(4X + 1)² and V(3 + 2X). Two fair dice are rolled. Let X be the maximum of the results of two dice. (a) Probability Mass Function (b) E(X) (c) V(X).

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

What is the covariance for the joint distribution below right? Cov(X, Y) = -1 YO +1 -1 1/16 3/16 1/16 X 0 3/16 0 3/16 +1 1/16 3/16 1/16