Find (a) Marginal density f X2 (b) Conditional density of the first, given that the second takes on the value e.
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- Let f (x) = Ke¯a (1 – e a"), x > 0. a. Find K such that f is a density. b. Find the corresponding distribution function. c. Find P (X > 1).Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature (°F) at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint density f(x, y) = = { x ² + 30.10. +,0Suppose that the joint distribution function of X and Y is given by f (x, y) =x+y If 0 < x < 1 and 0Only need part(d), (e), (f), thank you for your help!Consider times at which a parent provisions nestlings, that is, the time be- tween arrivals of a parent carrying food for the nestlings. Suppose that the waiting time between arrivals of a parent bird follows the exponential density function f (t) = 0.6e-0,61 for t z 0 and f (t) = 0 for t < 0, where the waiting times are in minutes. (a) What is the mean waiting time? (b) Find the probability that the wait will be between 2 and 5 minutes. (c) Find the probability that the wait will be at least 3 minutes.Suppose that an automobile dealer pays an amount X (in thousands) for a used car and then sells it for an amount Y. Suppose that X and Y have the joint density fxy (x, y) = for 0 < x < y < 6. The dealer's expected profit (in thousands) from the sale is equal toX and Y re random losses with joint density funcTion 2C forocn<00 500000 An Insurance losses pays the total of the twX9 losses to a maximum and oey< lo0, 100 policy on the payment of 100 the expectic fpagment the insurer will make on this policy.Let A > 0. Suppose (X, Y) has joint probability density function given, for all x, y E R, by 3. fx,y(x, y) = { Ay exp(-y(x+ A)), if a 2 0 and y > 0 0, otherwise. (i) Calculate the marginal density of Y. (ii) Let y > 0. Find the the conditional probability density function of X given Y = y. Interpret your answer. (iii) Calculate E[X|Y] and prove that E[X] = +oo. Hint: You can use without a proof that -dy = +oo. %3D#2, Suppose that X and Y have joint density given by: f (x, y) = 15x²y, 0 < x < y < 1. (a) Find the marginal pdf of Y. (b) Find the conditional pdf of X given Y. (c) Find the conditional expectation of X given Y = 1/3. (d) Are X andY independent? Show this in two different ways. (e) Give an integral expression for P(X + 2Y < 1), but do not evaluate it.answer quicklyXavier and Yolanda both have classes that end at noon and they agree to meet every day after class. They arrive at the coffee shop independently. Xavier's arrival time is X and Yolanda's arrival time is Y, where X and Y are measured in minutes after noon. The individual density functions are given.(Xavier arrives sometime after noon and is more likely to arrive promptly than late. Yolanda always arrives by 12:10 PM and is more likely to arrive late than promptly.) After Yolanda arrives, she'll wait for up to 45 minutes for Xavier, but he won't wait for her. Find the probability that they meet. (Round your answer to three decimal places.)Consider the probability density fy (x) = a . e~b lx!, where X is random variable whose allowable values range from x = - ∞ to + o. Find (a) the CDF (b) the relation between a and b (c) the probability that x lies between 1 and 2.Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON