Homework Answers
Let, W=X+Y
![c.d.f of W: Fu (w)=P(X+Y 5w) 55 P(x+Y5W) X=2) fy(x)dz [Law of total probability] :P($w.x) f(x) dx 0151646wdy] fm(z)dx [FCw-2)](http://img.homeworklib.com/questions/422752c0-73b4-11ea-9cb5-a15006a3282e.png?x-oss-process=image/resize,w_560)
![2221 ,0<x< 2 + 2y ,034 <i 3 fx(e) s fiery (35) dy = $5 (224) dy = $ [2+241 ${2+1] .ocza fy () = terpersdz = $ $(x+2y)dx= $ 4](http://img.homeworklib.com/questions/429f8420-73b4-11ea-a18c-29f64b79a4b1.png?x-oss-process=image/resize,w_560)
![1 | 2 (พ.-1)+2+ + 2 ผู-x)] 42 4 [ 24 - - - - 24 ] = 1 [ 4 * 15 46 ] [ 3 ] นะ -*)15+2 (1-2)42 =312v 941-- 211--11-1-4-)] : 1 (](http://img.homeworklib.com/questions/431cb2f0-73b4-11ea-adf7-511c2f60ea67.png?x-oss-process=image/resize,w_560)
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6. (10 points) Suppose X and Y are not independent, and are given by joint density...
Suppose that the joint probability density function of (X, Y) is given by fx, xx, y)...
Suppose that the joint probability density function of (X, Y) is given by fx, xx, y) =[1 – (1 – 2x)(1 – 2y)]lo (x), 1V), where the parameter o satisfies -1 Sa<1. (a) Prove or disprove: X and Y are independent if and only if X and Y are un- correlated. (X An isosceles triangle is formed as indicated in the sketch.
1. (10) Suppose the random variables X and Y have the joint probability density function 4x...
1. (10) Suppose the random variables X and Y have the joint probability density function 4x 2y f(x, y) for 0 x<3 and 0 < y < x +1 75 a) Determine the marginal probability density function of X. (6 pts) b) Determine the conditional probability of Y given X = 1. (4 pts)
1. Suppose the joint density of X and Y is given by f(x,y) = 6e-3x-2y, if...
1. Suppose the joint density of X and Y is given by f(x,y) = 6e-3x-2y, if 0 < x < inf., 0 < y < inf, 0 elsewhere. Part A, Find P( X < 2Y) Part B, Find Cov(X,Y) Part C, Suppose X and Y have joint density given by f(x,y) = 24xy, when 0<= x <=1, 0 <= y <=1, 0 <= x+y <=1, and 0 elsewhere. Are X and Y independent or dependent random variables? why?
Page (7) (10 points) The joint probability density function of X and Y is given by...
Page (7) (10 points) The joint probability density function of X and Y is given by a) Compute the marginal densities x and f b) Are X and Y independent? Why or why not? c) Compute P(Y > X7). MacBook Pro
3. Consider two random variables X and Y, whose joint density function is given as follows....
3. Consider two random variables X and Y, whose joint density function is given as follows. Let T be the triangle with vertices (0,0), (2,0), and (0,1). Then if (x, y for some constant K (a) (2 pts.) Find the constant K (b) (4 pts.) Find P(X +Y< 1) and P(X > Y). (c) (4 pts.) Find the marginal densities fx and fy. Conclude that X and Y are not independent
4. The random variables X and Y have joint probability density function fx,y(x, ) given by: fx,y(...
4. The random variables X and Y have joint probability density function fx,y(x, ) given by: fx,y(x, y) 0, else (a) Find c. (b) Find fx(x) and fy (), the marginal probability density functions of X and Y, respectively (c) Find fxjy (xly), the conditional probability density function of X given Y. For your limits (which you should not forget!), put y between constant bounds and then give the limits for in terms of y. (d) Are X and Y...
Suppose X and Y have joint distribution function given by: p(x, y) = for (x, y)...
Suppose X and Y have joint distribution function given by: p(x, y) = for (x, y) = (-1,0), (0,1), (1,0). (a) Are X and Y independent? (b) Find the Covariance of X and Y.
Q.4 (22') Suppose the joint probability density function of X and Y is fx,y(x, y) =...
Q.4 (22') Suppose the joint probability density function of X and Y is fx,y(x, y) = { „) - k(2 - x + y)x 0 sxs 1,0 sys1 o otherwise (a) (7”) Show that the value of constant k = 12 (b) (7') Find the marginal density function of X, i.e., fx(x). (c) (8') Find the conditional probability density of X given Y=y, i.e., fxy(xly). 11
a. Suppose X and Y are continuous random variables with joint denisty f(x,y). Prove that the...
a. Suppose X and Y are continuous random variables with joint
denisty f(x,y). Prove that the density of X+Y is given by:
Use part (a) to show that if X,Y are independent and standard
Gauss-ian (i.e.N(0,1)) then X+Yi s centered Gaussian with variance
2 that is N(0,2).
fx+r(t) = { $(8,6 – u)dt
The joint density function of continuous variables X and Y is (8 points) fry (x, y)...
The joint density function of continuous variables X and Y is (8 points) fry (x, y) = x y ; 0 < x < 1, 1 < y < 5 and= 0 elsewhere. i. Find the marginal density functions for X and Y, fx (x), fy (y). ii. Are X and Y independent?. Justify your answer.
Suppose that the joint probability density function of (X, Y) is given by fx, xx, y) =[1 – (1 – 2x)(1 – 2y)]lo (x), 1V), where the parameter o satisfies -1 Sa<1. (a) Prove or disprove: X and Y are independent if and only if X and Y are un- correlated. (X An isosceles triangle is formed as indicated in the sketch.
1. (10) Suppose the random variables X and Y have the joint probability density function 4x 2y f(x, y) for 0 x<3 and 0 < y < x +1 75 a) Determine the marginal probability density function of X. (6 pts) b) Determine the conditional probability of Y given X = 1. (4 pts)
Page (7) (10 points) The joint probability density function of X and Y is given by a) Compute the marginal densities x and f b) Are X and Y independent? Why or why not? c) Compute P(Y > X7). MacBook Pro
3. Consider two random variables X and Y, whose joint density function is given as follows. Let T be the triangle with vertices (0,0), (2,0), and (0,1). Then if (x, y for some constant K (a) (2 pts.) Find the constant K (b) (4 pts.) Find P(X +Y< 1) and P(X > Y). (c) (4 pts.) Find the marginal densities fx and fy. Conclude that X and Y are not independent
4. The random variables X and Y have joint probability density function fx,y(x, ) given by: fx,y(x, y) 0, else (a) Find c. (b) Find fx(x) and fy (), the marginal probability density functions of X and Y, respectively (c) Find fxjy (xly), the conditional probability density function of X given Y. For your limits (which you should not forget!), put y between constant bounds and then give the limits for in terms of y. (d) Are X and Y...
Suppose X and Y have joint distribution function given by: p(x, y) = for (x, y) = (-1,0), (0,1), (1,0). (a) Are X and Y independent? (b) Find the Covariance of X and Y.
Q.4 (22') Suppose the joint probability density function of X and Y is fx,y(x, y) = { „) - k(2 - x + y)x 0 sxs 1,0 sys1 o otherwise (a) (7”) Show that the value of constant k = 12 (b) (7') Find the marginal density function of X, i.e., fx(x). (c) (8') Find the conditional probability density of X given Y=y, i.e., fxy(xly). 11
a. Suppose X and Y are continuous random variables with joint
denisty f(x,y). Prove that the density of X+Y is given by:
Use part (a) to show that if X,Y are independent and standard
Gauss-ian (i.e.N(0,1)) then X+Yi s centered Gaussian with variance
2 that is N(0,2).
fx+r(t) = { $(8,6 – u)dt
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