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

a.)Moment Generating Function of U Define of the Moment Generating Function: The MGF of a random…

Answered: Question 3 [10 marks]. Suppose that X,…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 22E

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The table gives the joint probability distribution of the number of sports an individual plays (X) and the number of times she may get injured while playing (Y) X=1 X=2 X=3 0.12 0.08 0.15 0.06 0.05 0.05 0.10 0.03 0.15 0.15 0.04 0.02 Y=4 Y=3 Y=2 Y=1 The covariance between X and Y, oxy, is (Round your answer to two decimal places Enter a minus sign if your answer is negative) The correlation between X and Y, corr(X, Y), is (Round your answer to two decimal places Enter a minus sign if your answer is negative.) An increase in the number of sports an individual plays will tend to * the number of times she may got injured while playing
Customer A and B are being served by two clerks in the post office. Customer C is waiting and he is the last customer of today. Suppose a clerk’s serving time for any customer follows exponential distribution with parameter λ independently. Customers will be served once any of the clerks are available. What’s the probability that customer A the last customer to leave the post office?
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?
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?
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.
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
pls help
Let X be a Poisson random variable with E(X) = 3. Find P(2 < x < 4).
Let X be a discrete random variable with P(X = -1) = .3, P(X = 1) = .45, P(X = 2) = .15 and P(X = 6) = .1. Find Var(X).

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