Question

In: Statistics and Probability

Let (X, Y) be a random vector with a function of the joint density given by...

Let (X, Y) be a random vector with a function of the joint density given by ˜
fX, Y (x, y) = k (2x + y) I (2,6) (x) I (0.5) (y)

a) Determine k so that f X, Y (x, y) is a true probability density function joint quality.

b) Determine the marginal probability density functions of X and Y.

c) Calculate P (3 <X <4, Y> 2).

d) Calculate P (X + Y> 4).

Solutions

Expert Solution

please note that I(0.5) is improper representation for defining range, so assuming the correction to be as I(0,5) the question has been solved.

Thanks!


Related Solutions

Let (X, Y) be a random vector with a function of the joint density given by...
Let (X, Y) be a random vector with a function of the joint density given by ˜ fX, Y (x, y) = k (2x + y) I (2,6) (x) I (0.5) (y) a) Determine k so that f X, Y (x, y) is a true probability density function joint quality. b) Determine the marginal probability density functions of X and Y. c) Calculate P (3 <X <4, Y> 2). d) Calculate P (X + Y> 4).
. Let X and Y be a random variables with the joint probability density function fX,Y...
. Let X and Y be a random variables with the joint probability density function fX,Y (x, y) = { 1, 0 < x, y < 1 0, otherwise } . a. Let W = max(X, Y ) Compute the probability density function of W. b. Let U = min(X, Y ) Compute the probability density function of U. c. Compute the probability density function of X + Y ..
If the joint probability density function of the random variables X and Y is given by...
If the joint probability density function of the random variables X and Y is given by f(x, y) = (1/4)(x + 2y) for 0 < x < 2, 0 < y < 1, 0 elsewhere (a) Find the conditional density of Y given X = x, and use it to evaluate P (X + Y/2 ≥ 1 | X = 1/2) (b) Find the conditional mean and the conditional variance of Y given X = 1/2 (c) Find the variance...
Consider a continuous random vector (Y, X) with joint probability density function f(x, y) = 1...
Consider a continuous random vector (Y, X) with joint probability density function f(x, y) = 1                            for 0 < x < 1, x < y < x + 1. What is the marginal density of X and Y? Use this to compute Var(X) and Var(Y) Compute the expectation E[XY] Use the previous results to compute the correlation Corr (Y, X) Compute the third moment of Y, i.e., E[Y3]
Consider a continuous random vector (Y, X) with joint probability density function f(x, y) = 1...
Consider a continuous random vector (Y, X) with joint probability density function f(x, y) = 1 for 0 < x < 1, x < y < x + 1. A. What is the marginal density of X and Y ? Use this to compute Var(X) and Var(Y). B. Compute the expectation E[XY] C. Use the previous results to compute the correlation Corr(Y, X). D. Compute the third moment of Y , i.e., E[Y3].
Let X and Y be two continuous random variables with the joint probability density function of...
Let X and Y be two continuous random variables with the joint probability density function of for 0 < x < 2, 0 < y < 2, x + y < 1,where c is a constant. (In all the following answers, you do NOT need to find what the value of c is; just treat it as a number.) (a) Write out the marginal distribution of Y. (b) P(Y < 1/3) = ? (c) P(X < 1.5, Y < 0.5)=...
. The joint probability density function of X and Y is given by ?(?, ?) =...
. The joint probability density function of X and Y is given by ?(?, ?) = { ??^2? ?? 0 ≤ ? ≤ 2, 0 ≤ ?, ??? ? + ? ≤ 1 0 ??ℎ?????? (a) Determine the value of c. (b) Find the marginal probability density function of X and Y. (c) Compute ???(?, ?). (d) Compute ???(?^2 + ?). (e) Determine if X and Y are independent
The joint probability density function for two random variables X and Y is given as, fx,y...
The joint probability density function for two random variables X and Y is given as, fx,y (x, y) = (2/3)(1 + 2xy3 ), 0 < x < 1, 0 < y < 1 (a) Find the marginal probability density functions for X and Y . (b) Are X and Y independent? Justify your answer. (c) Show that E[X] = 4/9 and E[Y ] = 7/15 . (d) Calculate Cov(X, Y )
The joint probability density function (PDF) of two random variables (X,Y) is given by ???(?,?) =...
The joint probability density function (PDF) of two random variables (X,Y) is given by ???(?,?) = { 1, 0 ≤ ? ≤ 2,0 ≤ ? ≤ 1,2? ≤ ? 0, otherwise 1) Find the correlation coefficient ??? between the two random variables X and Y Find the probability P(Y>X/2). help please asap
Suppose that the joint probability density function of ˜ (X, Y) is given by:´ f X,Y...
Suppose that the joint probability density function of ˜ (X, Y) is given by:´ f X,Y (x,y) = 4x/y3 I(0.1)(x), I (1, ∞)(y). Calculate a) P(1/2 < X < 3/4, 0 < Y ≤ 1/3). b) P(Y > 5). c) P(Y > X).
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT