4.144 Consider a random variable Y with density function given by f(y)=ke-²/2 a Find k. b Find the moment-generating function of Y. c Find E(Y) and V (Y). -∞
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- Let X and Y be independent Gaussian random variables, each distributed according to (0, σ2). 1. Find the joint density function of the random variables Z = X + Y and W = 2X −Y . What is the correlation coefficient between these two random variablesSuppose that Y1, . . . , Yn is a random sample from a population whose density function is7.1 Let X be a random variable with probability Si. 1= 1,2, 3, S(2) =10 elsewhere. Find the probability distribution of the random vari- able Y = 2X 1. 7.8 A dealer's profit, in units of $5000, on a new au- tomobile is given by Y = X², where X variable having the density function sa random S(z) =2(1 - 2), 02.5.5Suppose that two continuous random variables X and Y have a joint probability density function f(x, y) = A(ex+y + 2x-y) for 1 ≤ x ≤ 2 and 0 ≤ y ≤ 3, and f(x, y) = 0 elsewhere. (a) What is the value of A? (b) What is P(1.5 = X = 2, 1 = Y = 2)? (c) Construct the marginal probability density functions fx(x) and fy(y). (d) Are the random variables X and Yindependent? (e) If Y= 0, what is the conditional probability density function of X?1. Let X be a random number from (0, 1). Find the probability density function of Y = 1/X.Suppose that X is a random variable whose density function is defined as follows: (a+1) 2a fx(x) = 2a+1 with 0 -1. For a = 1.15 calculate the IQR of X.2. Let X be a Chi-squared random variable with density function fx(x) = exp x > 0. 2 Suppose Y is independent of, and has the same distribution as, X.6.74 Let Y₁, ₂, ..., Y, be independent, uniformly distributed random variables on the interval [0, 0]. Find the a probability distribution function of Y() = max(Y₁; Y₂;; Y₁). b density function of y(n). c mean and variance of y(n).7. A random variable X has probability density function (pdf) fx (u) = |u+ 1| for u E -2,0] and fx(u) = 0 otherwise. (a) Find the variance of X. (b) Calculate the CDF Fx(u) for-oo < u < 0o and sketch a plot of it.10. Suppose a random variable X has an exponential distribution with A = 1. Let Y = X². (a) Derive a formula for the cdf of Y. (b) Find the density of Y using part (a).2. Y1, Y2, ..., Yn are i.i.d. exponential random variables with E{Yi} = 1/θ. Find thedistribution of Y =1 nPiYi.5. Let 0 and a be parameters. Show that the family of functions fo,a(x) = I > 0 (x+0)a+1³ is a probability density function. What can you say about the mean of a random variable that has fo,a as a pdf?SEE MORE QUESTIONS