Problem 4: A company produces dyes for the food industry by mixing cyan, magenta, andyellow dyes. Let X and Y, respectively represent the fraction of cyan and magenta dye in oneorder of dye from one customer. The joint density of X and Y is given byf(x,y) = {24xy0 ≤ y ≤ 1, x + y ≤ 1,0 ≤ x ≤ 1,elsewhere.a) Find the probability density function of the random variable Z = X + Y.b) Use the density function of Z to find the probability that the fraction of yellow dye in anorder is at most 0.1.

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Answered: Problem 4: A company produces dyes for…

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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2. Let X and Y be random variables, and let a and b be constants. (a) Starting from the definition of covariance, show that Cov(aX,Y) = a Cov(X,Y). You may find it helpful to remember that if EX = µx, then EaX = aux. (b) Show that Cov(X + b, Y) = Cov(X,Y). Now let X, Y, Z be independent random variables with common variance o?. (c) Find the value of Corr(2X – 3Y + 4, 2Y – Z – 1). You may use any facts about covariance from - the notes, including those from parts (a) and (b) of this question, provided you state them clearly.
6. Three points A, B, C are chosen independently at random on the circumference of a circle. Let b(x) be the probability that at least one of the angles of the triangle ABC exceeds xл. Show that - 1 − (3x − 1)² if ⁄ ≤ x ≤ 1, if ≤ x ≤ 1. 3(1-x)² Hence find the density and expectation of the largest angle in the triangle. b(x) = {
You roll an 8-sided die. Let A = {4, 5, 7, 8} and B = {1, 3, 6, 7}. Find the following. S = { A' = B' = P(A') = P(B') = n(A' AND B) = n(A OR B') = P(A OR B') = Ρ(A' AND Β) - P(A | B') = %3D P(A' | B) = %3D
Let X1,..., X10 be a random sample of size 10 from a N(u, o²) population. Suppose 10 Y =(X; - 4)? i=1 Find the probability that the random interval Y Y 20.5' 3.25 includes the point o?.
2.Suppose that the return R (in dollars per share) of a stock has the uniform distribution on the interval [-3,7]. Suppose also, that each share of the stock costs $1.50. Let Y be the net return (total return minus cost) on an investment of 10 shares of the stocks. Compute E(Y).

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