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?.
Example 4-28. (Buffon's Needle Problem). A vertical board is ruled with horizontal parallel lines at constant distance 'a' apart. A needle of length l (< a) is thrown at random on the table. Find the probability that it will intersect one of the lines.
If X and Y are independent RVs, each following N (0, 3), what is the probability that the point (X, Y) lies between the lines 3X + 4Y = 5 and 3X + 4Y = 10?
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).
7. Given that f(x, y) = (2x+2y)/2k if x = 0,1 and y = 1,4, is a joint probability distribution function for the random variables X and Y. Find: The marginal function of x %3D
If λ = 6 per hour and μ = 6 per hour in an (M/M/1): (5/FIFO) model, find the probability that there is no customer in the system.
1. If the joint probability is given by: r + y f(x, y) for r = 0,1,2, 3; y = 0, 1,2, 30 Find: (a) P(Xs 2, Y = 1); (b) P(X > 2, Ys1); (c) P(X>Y ); (d) P(X + Y = 4). 2. From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4 pieces of fruit is selected. If X is the number of oranges and Y is the number of apples in the sample, find: (a) the joint probability distribution of X and Y ; (b) P[(X, Y ) E A), where A is the region that is given by {(x, y) | x + ys 2}.

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