Let (x,y) have pdf f(x,y) = 1 over the region R = triangle with vertices (0,0), (1,0), (0,2) and f(x,y) = 0 elsewhere. Define Z = 2X + Y, W = 2X. a) Inverse transformation from (z,w) to (x,y) b) Jacobian J = c) sketch the region R in (x,y) coordinates d) sketch the region D in (z,w) coordinates e) Joint pdf over D, g(z,w) = f) Find pdf h(z) of Z from marginal of g(z,w) and sketch the graph H1

Let (x,y) have pdf f(x,y) = 1 over the region R = triangle with vertices (0,0), (1,0), (0,2) and f(x,y) = 0 elsewhere. Define Z = 2X + Y, W = 2X. a) Inverse transformation from (z,w) to (x,y) b) Jacobian J = c) sketch the region R in (x,y) coordinates d) sketch the region D in (z,w) coordinates e) Joint pdf over D, g(z,w) = f) Find pdf h(z) of Z from marginal of g(z,w) and sketch the graph H1

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Let (x,y) have pdf f(x,y) = 1 over the region R = triangle with vertices (0,0), (1,0), (0,2) and f(x,y) = 0 elsewhere. Define Z = 2X + Y, W = 2X.

a) Inverse transformation from (z,w) to (x,y)

b) Jacobian J =

c) sketch the region R in (x,y) coordinates

d) sketch the region D in (z,w) coordinates

e) Joint pdf over D, g(z,w) =

f) Find pdf h(z) of Z from marginal of g(z,w) and sketch the graph

Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.

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