3 Let Y be a random variable with probability density function given by 2(1-y), Osys 1, fly) 0, elsewhere. Use the method of transformation to find the densities of U₁, U., and U. (a) U-2Y-3 (L) elsewhere (b) U₂-3-2 (3)- (c) U₁₂ = y2 fu₂ (4) - elsewhere elsewhere SUS SUS SUS
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- a. Explain the difference between the method of moment generating function and the method of incarnation to find the probability density function of a random variable b. Let X1, X2, X3 ,,,,, each of them has Poisson distribution with parameter lamda 1, lamda2, lamda3,........ i. Get the probability distribution of Y=X1+X2+....Xn ii. Get the mean and variance of Y c. Y1 and Y2 be a joint probability function as shown below in the screenshot imageU=Y1/(Y1+Y2), Using the method of incarnation (Jacobian), find the probability density function of Ua) Find the moment generating function MX(t) for a discrete random variable X that takes the value -1 with probability 1/2 and takes the value +1 with probability 1/2. b) Find the moment generating function MU(t) for the standard uniform random variable U (the continuous random variable whose density function is 1 on [0,1] and 0 elsewhere). c) Find the moment generating function ME(t) for an exponential random variable with parameter (lambda) = 1. Sketch the graph of ME(t).b) Let Y1, Y2, ..., Yn be a random sample from a population with probability density function in part a). Show that the best test for the hypothesis in part a) rejects Ho if |yi <c i=1 where c solves the probability equation a = P(II1Yis c[0 = 2). c) Let X1,X2, ..., Xn be a random sample from GAMMA(2,ß) distribution, and consider Y = E-,Xi- Show whether or not Y is a pivotal quantity and give its distribution.
- X is a uniform random variable over the interval (3, 5). Find the density function of X for the interval (3, 5)An electronic device contains two circuits. The second circuit is a backup for the first and is switched on only when the first circuit has failed. The electronic device goes down when the second circuit fails. The continuous random variables X and Y denote the lifetimes of the first circuit and the second circuit and have the joint density function f(x, y) = 24/(x + y)4 for x, y > 1 and f(x, y) = 0 otherwise. What is the expected value of the time until the electronic device goes down? What is the probability density function of this time?Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.
- 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 variablesLet X be a continuous random variable with probability density function f(x) = 2(1-x), 0<=x <= 1. If Y = 2X - 1, find the probability density function of Y. x is lower case X is upper case. I need to get upper case of YLet G be a geometric random variable with success probability p. Now, conditional on G = g, let X = U1U2 . . .Ug be a product of g independent and identically distributed uniform (on [0, 1]) random variables. For example, if g = 3 then X = U1U2U3 where the U1,U2,U3 are iid U[0, 1] random variables.Find the unconditional probability density function for X.
- Let X be a discrete random variable taking values {x1, x2, . . . , xn} with probability {p1, p2, . . . , pn}. The entropyof the random variable is defined as H(X) = −sigma(pilog(pi)) ( where sigma takes value i=1 to n ) Find the probability mass function for the above discrete random variable that maximizes the entropyLet x be a random variable with probability mass function given by: (image) Find V[X]Let X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.