Suppose you have joint random variables X and Y.a) If a problem does not state anything about whether the variables are independent or not, can we assumeit?YesO Nob) How would you calculate Cov(X,Y) if we have no information about independence?О Е (XҮ) — Е(Хх)E(Y) — 0Ο Ε(ΧY)-E(X)E(Υ)c) If random variables X and Y are independent, how would you calculate Cov(X, Y)?Ο Ε(XY)-E(X)Ε(Υ) 0О Е (XҮ) — Е(X)E(Y)

X and Y are 2 R.V. and if the dist. of X is not influenced by Y values then 2 R.V. are independent.

Answered: Suppose you have joint random variables…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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2. If X is a random variable taking values 0,1,2,.... and P(X) = ab*, where a and b are positive constants such that a+b= 1, then (i) Find mgf. (ii) If E(X) = m₁, E(X2) = m2, show that m₂ = m₁(2m₁ + 1).
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Let X1, X2, X3 be random variables each having a mean u and variance o. Further, Cov(X1, X2) = 2, Cov(X1.X3) = 3 and Cov(X2, X3) = 1 Define U = 2X1 – X2 + 4 X3 Solve for the mean and standard deviation of U.
c. Suppose that the random variables X1, X2 and X3 are independent with means and variances as follows: H1 = 8 of = 36 H2 = 5 o = 81 H3 = 7 o3 = 49 Let Y1 = X1 + 2X, and Y2 = 2X1 – }X2 – X3. (i) Find the Cov(Y1, Y2). (ii) Assuming that X1, X2 and X3 are Normal random variables, calculate P(Y < 2Y,).
2) What can be said about the correlation coefficient r of x and y where y = x² + 8x + 16, a takes the values of the first ten positive integers? r = 1 O 0
2. The joint PDF of two random variables X and Y is given by ху fxy (x, y) (x2 +).0 Y] c) Find Pr[Y >1X <
18. Suppose X and Y are random variables with E (X) = 2 E (Y) = 3 E (X*) = 53 E (Y*) = 45 E (XY) = -21 (a) Find V (X) (b) Find V (Y) (c) Find Cov(X, Y) (d) Find V (2X –- 3Y')
Suppose that X, Y, and Z are random variables with X - Bin(16, 1/4), Y ~ Geom(1/2), and Z ~ Poisson(5). Suppose further that X and Z are independent but that X and Y are not independent. Match up the following quantities. E(4Y – 2Z) Drag answer here 23 Var(2X + Y) 8 Drag answer here Var(X – Z) -2 Drag answer here E(Y² + Z®) Cannot be determined Drag answer here 36 Var(X+ 2Z) Drag answer here
3. If X;~N(0,1), i = 1,2,3,4 are independent Normal random variables and Y is given as Y =(X,+X,)' +(X, + X,)' . If aY~x², . find the constant a. 2
Let X be a discrete random variable with the following PMF 1 1 for x=-2 8 100 for x = -1 Px(x) = for x = 0 1 for x = 1 for x = 2 4 0 otherwise a) Find E[X] and Var(X) b) Now we define a new random variableY = (x+1)², Find PMF of Y and E[Y]
4 Suppose that Alejandro is planting a garden of tulips. Let X be the Bernoulli random variable that returns 1 if the tulip is red, and 0 otherwise. Suppose Y is another Bernoulli random variable, independent of X, that returns 1 if the tulip survives longer than 30 days, and 0 otherwise. Both X and Y have parameters 1/2. Let Z be the random variable that returns the remainder of the division of X +Y by 2. For example, if X = 1 and Y = 0, Z = remainder() = 1 (a) Prove that Z is also a Bernoulli random variable, also with parameter 1/2. (b) Prove that X, Y, Z are pairwise independent but not mutually independent. (c) By computing Var[X+Y+Z] according to the alternative formula for variance and using the variance of Bernoulli r.v.'s, verify that Var[X+Y+Z] = Var[X]+Var[Y] +Var[Z] %3D (observe that this also follows from the proposition on slide 5 of the lecture segment entitled "Binomial distribution").
Suppose we choose arbitrarily a point from the square with corners at(2,1), (3,1), (2,2), and (3,2). The random variable A is the area of the trianglewith its corners at (2,1), (3,1), and the chosen point. Compute E[A].

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