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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).Suppose the joint probability density function of X and Y is given by 3 (1- Vx² + y² ) for ² + y? < 1 for a? + y? < 1 f(x, y) otherwise In other words, f(x, y) takes non-zero values when (r, y) is inside the unit circle. Calculate E[XY].1. Let X be a continuous random variable with the following density function: cx,0Let X and Y be continuous random variables with joint p.d. f. f(x,y) = {* (12x for 0< y< 2x < 1 elsewhere The conditional density function of Y given X = x isConsider the continuous density function f(x) = 32 (x+4)³ , defined for x > 0. 7 a) Construct the cumulative distribution function for the above density function. b) Find P(1 < X < 4)Let X have the probability density function given by Se-*, 0A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X be the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is fo fkr², if 0The random variable X has probability density function (ke , x > 10 20 2 f (x) = 0. otherwise where k is a constant. Find the value of k that makes f a valid probabilty density function.5. If the joint probability density function of X and Y is f (x, y) = { ! what X and Y ? OLXLI, 0 Ly Lit otherwise. 25100 moment generating function of is the joint momentRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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