3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X). b) Show that Z=-2ln(Y) has a Gamma dist. & derive it.
4. X_i ~ cont with pdf f_i(x) and CDF F_i(x), i=1, 2, ..., k. all independent. Define Y_i=F_i(X_i), i=1, ..., k. Derive the distribution of U=-2ln[Y_1.Y_2...Y_k].
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist...
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X) b) Show that Z=-21n(Y) has a Gamma dist. & derive it. 4. X-i ~ cont with pdf fi(x) and CDF Fi(x), i=1, 2, , k. all independent. Define YjaFi(Xi), i=1, , k. Derive the distribution of 3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X) b) Show that Z=-21n(Y) has a Gamma dist....
Let Y be some rv (discrete or continuous). Let the transformation be: U = F (Y ), where F (·) stands for the cdf. Find the pdf of U using the cdf method. Can you name the distribution of U?
(a) Let X be a continuous random variable with the cdf F(x) and pdf f(.1). Find the cdf and pdf of |X|. (b) Let Z ~ N(0,1), find the cdf and pdf of |Z| (express the cdf using ” (-), the cdf of Z; give the explicit formula for the pdf).
Problem 2. Assume a random vector (X Y with cdf F(r, ) and pdf f(r,y) (i) Show that Y/X has the pdf f(x, z) |da, g(z) = (ii) For X and identify the distribution of this pdf. xt independent, evaluate the pdf of Y/VX N(0, 1) and Y
2. Let X be a continuous r.v. with pdf f () and cdf F(x). Let U F (X). Show that, as long as F(x) is strictly monotonic increasing, U is uniformly distributed on (0,1). Discuss why this result is important, given that it is known how to simulate Uniformly distributed random variables easily.
7.1 required non-book problem: Suppose RV Y is continuous with invertible CDF Fy. Then 1. U = Fy (Y) is uniform on the unit interval, i.e., U U (0,1). Recall that this result is known as the Probability Integral Transform. 2. Y = F'(U) has CDF Fy if U U (0,1). Do the following: 1: Let W = Fy(Y) where Fy(y) = 1 - e-dy, osy< where Y is exponential with parameter and Fy is the CDF of Y. Using...
Show all work Question # A.1 (a) Given the CDF of a RV. x is specified as follows: Fx(x) = = = 0 B x (x/3) 1 in the range (x<0) in the range (0<x<+1) in the range (x > 1) Determine the following: (i) Value of B: (i) pdf of x: (111) pdf of y under the transformation y = (ax + b) where a and b are constants; (iv) range of the transformed RV, y and sketch the...
1. Suppose that X is continuous random variable with PDF f(x) and CDF F(x). (a) Prove that if f(x) > 0 only on a single (possible infinite) interval of the real numbers then F(x) is a strictly increasing function of x over that interval. [Hint: Try proof by contradiction]. (b) Under the conditions described in part (a), find and identify the distribution of Y = F(x). 2. Suppose now that X ~ Uniform(0, 1). For each of the distributions listed...
5. (20 pts) Function of RV Let Ry X-Exponential(1),i.e.,the CDF is Fx (x) = (1 - )u(x). IEX = 9(x) = -2x + 1, find the CDF Fy (y) and the PDF fy(y).
P7 continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...