3. Let f be the density function of a random variable X, and let Y = (Y1, Y2, Y3) be a continuousrandom vector with density function%3!fY (y1, V2, ys) 6 f(v)f(v)/() 1{vn < ya < ya}.(a) Find the conditional density fy, Y,Y, (V, Vsly2) and argue that (Y I. Y) | Y.(b) If X has a uniform distribution on the unit interval (0, 1], deduce that (Y, Y) | (Y = V)has a certain bivariate uniform distribution. Over which set?

This question involves working with random variables and their density functions. Specifically, we…

Answered: 3. Let f be the density function of a…

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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9. The density of the random vector (X, Y)' is cx, for x > 0, y 2 0, x +y <1, fx,y (x, y) otherwise. Compute (а) с, (b) the conditional expectations E(Y | X = x) and E(X |Y = y).
Random variables X and Y are independent and distributed as Uniform(−1,3).Find the density of random vector (Z,W), say f(z,w), where Z and W are defined asZ = X + 2Y, W = 2X −Y.
6. Let f(x1,x2) = 1²e¬Mx1+*2), Let Y1 = X1 + X2 and X2 = Y2. (a) Find the joint distribution of Y1 and Y2: f(y1, y2) (b) Then compute the marginal density of Y1
The random vector (X,Y) has the following joint probability density function:f(X,Y)(x,y) ={4xye−(x^2+y^2), x >0, y >0, 0, otherwise LetZ=√(X2+Y2) Find the probability density of the random variableZ.
1)A random vector z = (x y) T has joint probability density given by:a)Sketch the graph of the probability density function pxy(X, Y ).b)Determine the value of the constant C.c)Determine the mean mz vector.
7. Let X and Y denote two continuous random variables. Let f(x,y) denote the joint probability density function and fx(x) and fy (y) the marginal probability density functions for X and Y, respectively. Finally let Z = aX + bY, where a and b are non-zero real numbers. (e) Derive an expression for Cov(Z) as a function of Var (X), Var(Y) and Cov(X,Y). [You may use standard results relating to variance and covariance without proof, but these should be clearly stated.]
Exercise 40 Let X Unif(0,1). Let g(x) = e" and Y = g(X). (i) Find the density fx of the random variable X. (ii) Find the cdf Fy (y) = P(Y < y) of Y. (iii) Find the density fy of the random variable Y.
A system of random variables (X, Y) is normally distri- buted with the probability density ƒ(x, y) = 21/02 exp{-*² 2 +2 1². 20² Find the probability density of the system (R, D) if X = R cos , Y = R sin Þ.
2. Let X and Y denote independent random variables with respective probability density func- tions fx(x) = 2x, 0
3. Suppose that X and Y are independent with X ~ N(0, 1) and Y~ N(0,4). Let U = X + Y and V = X - Y. a) Find the joint probability density function of (U, V), fu,v(u, v). b) Are U and V independent?
Choose a number B at random from the interval [0, 1] with uniform density. Find the probability that (a) 1/3 < B < 2/3(b) |B − 1/2| ≤ 1/4(c) B < 1/4 or 1 − B < 1/4(d) 3B2 < B
X and Y are independent variables, which distribution is given by is given by the functions: (picture) Determine the density of random vector (X,Y)' Determine F(5,2) Determine P(Y>2X) and sketch the picture

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