x | |||
y | -1 | 1 | Total |
-1 | 3/8 | 1/8 | 1/2 |
1 | 1/8 | 3/8 | 1/2 |
Total | 1/2 | 1/2 | 1 |
a)
marginal distribution of X:
x | P(x) |
-1 | 1/2 |
1 | 1/2 |
marginal distribution of Y:
y | P(y) |
-1 | 1/2 |
1 | 1/2 |
b)
below is condition distribution of X given Y=1
P(X=-1|Y=1)=(1/8)/(1/2)=1/4
P(X=1|Y=1)=(3/8)/(1/2)=3/4
c)
No as P(X|Y=1) is different for X =-1,1 therefore X and Y are not independent
d)
x | P(x) | xP(x) | x^2P(x) |
-1 | 1/2 | -0.5000 | 0.5000 |
1 | 1/2 | 0.5000 | 0.5000 |
total | 1.0000 | 0.0000 | 1.0000 |
E(x) | = | 0.0000 | |
E(x^2) | = | 1.0000 | |
Var(x) | E(x^2)-(E(x))^2 | 1.0000 |
from above E(X)=0.00
Var(X)=1.00
y | P(y) | yP(y) | y^2P(y) |
-1 | 1/2 | -0.5000 | 0.5000 |
1 | 1/2 | 0.5000 | 0.5000 |
total | 1 | 0 | 1 |
E(y) | = | 0.0000 | |
E(y^2) | = | 1.0000 | |
Var(y) | E(y^2)-(E(y))^2 | 1.0000 |
E(Y)=0.00
Var(Y)=1.00
e)
P(XY<0)=P(X=-1,Y=1)+P(X=1,Y=-1)=1/8+1/8=1/4
P(max(X,Y)>0)=1-P(max(X,Y)<0)=1-P(X=-1,Y=-1)=1-3/8=5/8
f)
E(max(X,Y))=max(X,Y)*P(x,y)=(3/8)*max(-1,-1)+(1/8)*(1,-1)+(3/8)*max(1,1)+(1/8)*(-1,1)
=2/8=1/4
EXY)=xy*P(x,y)=0.5
g)
Cov(X,Y)=E(XY)-E(X)*E(Y)=0.50
Corr(X,Y)= Cov(X,Y)/sqrt(Var(X)*Var(Y))=0.50
1. Let the joint probability (mass) function of X and Y be given by the following:...
Question 4: Let X and Y be two discrete random variables with the following joint probability distribution (mass) function Pxy(x, y): a) Complete the following probability table: Y 2 f(x)=P(X=x) 1 3 4 0 0 0.08 0.06 0.05 0.02 0.07 0.08 0.06 0.12 0.05 0.03 0.06 0.05 0.04 0.03 0.01 0.02 0.03 0.04 2 3 foy)=P(Y=y) 0.03 b) What is P(X s 2 and YS 3)? c) Find the marginal probability distribution (mass) function of X; [f(x)]. d) Find the...
(20 points) Consider the following joint distribution of X and Y ㄨㄧㄚ 0 0.1 0.2 1 0.3 0.4 (a) Find the marginal distributions of X and Y. (i.e., Px(x) and Py()) (b) Find the conditional distribution of X given Y-0. (i.e., Pxjy (xY-0)) (c) Compute EXIY-01 and Var(X)Y = 0). (d) Find the covariance between X and Y. (i.e., Cov(X, Y)) (e) Are X and Y independent? Justify your answer. (20 points) Consider the following joint distribution of X and...
1. The joint probability density function (pdf) of X and Y is given by fxy(x, y) = A (1 – xey, 0<x<1,0 < y < 0 (a) Find the constant A. (b) Find the marginal pdfs of X and Y. (c) Find E(X) and E(Y). (d) Find E(XY). 2. Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day. Let Y denote the number of times (1, 2, or 3...
3. Let the random variables X and Y have the joint probability density function 0 y 1, 0 x < y fxy(x, y)y otherwise (a) Compute the joint expectation E(XY) (b) Compute the marginal expectations E(X) and E (Y) (c) Compute the covariance Cov(X, Y)
Let X and Y be jointly continuous random variables with joint probability density given by f(x, y) = 12/5(2x − x2 − xy) for 0 < x < 1, 0 < y < 1 0 otherwise (a) Find the marginal densities for X and Y . (b) Find the conditional density for X given Y = y and the conditional density for Y given X = x. (c) Compute the probability P(1/2 < X < 1|Y =1/4). (d) Determine whether...
Let X and Y be jointly continuous random variables with joint probability density given by f(x, y) = 12/5(2x − x2 − xy) for 0 < x < 1, 0 < y < 1 0 otherwise (a) Find the marginal densities for X and Y . (b) Find the conditional density for X given Y = y and the conditional density for Y given X = x. (c) Compute the probability P(1/2 < X < 1|Y =1/4). (d) Determine whether...
Let the frequency function of the joint distribution of the random variables X and Y P(X = 2, Y = 3) = P(X = 1, Y = 2) = P(X = -1, Y = 1) = P(X = 0, Y = -1) = P(X = -1, Y = -2) = 3 a) Determine the marginal distributions of the random variables X and Y. b) Determine Cov(X,Y) and Corr(X,Y). c) Determine the conditional distributions of the random variable Y as a...
Q3. . Suppose that joint probability function of X and Y is given by | 1/7, z = 5, y = 0 Px,y(, ) 0, otherwise. a. Find the marginal distribution of X and Y b. Find E(X|y = 4] c. Compute Cov(X, Y). d. Are X, Y independent? justify e. Compute E[XY0or4] f. Find px(8) and P(Y-4X-8).
3. Let the random variables X and Y have the joint probability density function fxr (x, y) = 0 <y<1, 0<xsy otherwise (a) Compute the joint expectation E(XY). (b) Compute the marginal expectations E(X) and E(Y). (c) Compute the covariance Cov(X,Y).
Please do not copy, all the previous answers are wrong. 3. The joint probability density function of X and Y is given by 2 if O< x S 2,0 < y, and x +ys1 otherwise f(x,y) = 〉cry (a) Determine the value of c (b) Find the marginal probability density function of X and Y (c) Compute Cov(X, Y) (d) Compute Var(X2 Y) (e) Determine if X and Y are independent. 3. The joint probability density function of X and...