Question

In: Statistics and Probability

X and Y are discrete random variables with joint distribution given below Y 1 Y 0...

X and Y are discrete random variables with joint distribution given below Y 1 Y 0 Y 1 X 1 0 1/4 0

X 0 1/4 1/4 1/4

(a) Determine the conditional expectation E Y|X 1 . (b) Determine the conditional expectation E X|Y y for each value of y. (c) Determine the expected value of X using conditional expectation results form part (b)

above. (d) Now obatin the marginal distribution of X and verify your answers to part (c).

Expert Solution

Answer:-

Given that:-

X and Y are discete random variables with joint distribution

	Y=-1	Y=0	Y=1	Total
X=1	0	1/4	0	1/4
X=0	1/4	1/4	1/4	3/4
Total	1/4	2/4	1/4	1

a) Determine condition Expectation E(y/x=1)

E(y/x=1)=

Condition probability distribution of y given x=1

Condition probability distribution of y/x=1

y	-1	0	1
p(y/x=1)	0	1	0	1

b)Detremine the condition expectation E(x/y=y) for each value or y

E(x/y=-1), E(x/y=0), E(x/y=1)

	Y=-1	Y=0	Y=1	Total
X=1	0	1/4	0	1/4
X=0	1/4	1/4	1/4	3/4
Total	1/4	2/4	1/4	1

To find E(x/y=-1)

Condition prob.distribution of x/y=-1

Condition prob distribution of x/y=-1

x	1	0
p(x/y=-1)	0	1	1

E(x/y=-1)=

E(x/y=-1)=0

To find E(x/y=0)

condition probability of x/y=0

condition prob distribution of x/y=0

x	1	0
p(x/y=0)	1/2	1/2	1

E(x/y=0)=

To find E(x/y=1)

Condition probability of x/y=1

condition prob distribution or x/y=1

x	1	0
p(x/y=1)	0	1	1

E(x/y=1)=

c)We know that from low of it created

expectation

E[E(x/y)]=E(x)

g(y)=E(x/y=y)

E[E(x/y)]=E(g(y))

=

E[E(x/y=y)]=E(x/y=-1).p(y=-1)+E(x/y=0)(y=0)+E(x/y=1).p(y=1)

E[E(x/y)]=

Therefore E(x)=

Therefore, Expected value of x using condition expectation from b]

d)To obtain marginal distribution of x.

	Y=-1	Y=0	Y=1	Total
X=1	0	1/4	0	1/4
X=0	1/4	1/4	1/4	3/4
Total	1/4	2/4	1/4	1

Marginal distribution of x

x	1	0
p(x)	1/4	3/4	1

E(x)=

Now from (c) and (d) we verify that

orchestra answered 1 year ago

The table below gives the joint distribution between random variables X and Y . y 0...

The table below gives the joint distribution between random variables X and Y . y 0 2 4 x 0 0.03 0.01 0.20 1 0.15 0.10 0.51 (a). [4pts] Find P(X = 0, Y = 2). 1 Problem 2(b). [6pts] Find E[X]. Problem 2(c). [6pts] Find Cov(X, Y ). Problem 2(d). [6pts] Find P(X = 1|Y = 2).

Let X and Y be two discrete random variables whose joint probability distribution function is given...

Let X and Y be two discrete random variables whose joint probability distribution function is given by f(x, y) = 1 12 x 2 + y for x = −1, 1 and y = 0, 1, 2 a) Find cov(X, Y ). b) Determine whether X and Y are independent or not. (Justify your answer.

.The following table displays the joint probability distribution of two discrete random variables X and Y....

.The following table displays the joint probability distribution of two discrete random variables X and Y. -1 0 1 2 1 0.2 0 0.16 0.12 0 0.3 0.12 0.1 0 What is P(X=1/Y=1)? What is the value of E(X/Y=1)? What is the value of VAR(X/Y = 1)? What is the correlation between X and Y? What is variance of W = 4X - 2Y. What is covariance between X and W?

Given below is a bivariate distribution for the random variables x and y. f(x, y) x...

Given below is a bivariate distribution for the random variables x and y. f(x, y) x y 0.3 50 80 0.2 30 50 0.5 40 60 (a) Compute the expected value and the variance for x and y. E(x) = E(y) = Var(x) = Var(y) = (b) Develop a probability distribution for x + y. x + y f(x + y) 130 80 100 (c) Using the result of part (b), compute E(x + y) and Var(x + y). E(x...

Suppose the joint probability distribution of two binary random variables X and Y are given as...

Suppose the joint probability distribution of two binary random variables X and Y are given as follows. x/y 1 2 0 3/10 0 1 4/10 3/10 X goes along side as 0 and 1, Y goes along top as 1 and 2 e) Find joint entropy H(X, Y ). f) Suppose X and Y are independent. Show that H(X|Y ) = H(X). g) Suppose X and Y are independent. Show that H(X, Y ) = H(X) + H(Y ). h)...

Discrete random variables X and Y have joint probability mass function defined by the table below....

Discrete random variables X and Y have joint probability mass function defined by the table below. X= { 0 , 1 , 2 } , Y= { − 2 , 0 , 2 } y/x 0 1 2 -2 0.2 0.1 0.1 0 0.0 0.4 0.0 2 0.1 0.0 0.1 A.) Evaluate the expected value of X (round to 2 decimal digits) B.) Evaluate the variance of X (round to 2 decimal digits) C.) Evaluate the covariance of X...

Define a joint distribution for two random variables (X,Y) such that (i) Cov(X,Y)=0 and (ii) E[Y...

Define a joint distribution for two random variables (X,Y) such that (i) Cov(X,Y)=0 and (ii) E[Y I X] is not equal to E[Y]. How do I define a joint distribution that satisfies both (i) and (ii) above at the same time? Please give me an example and explanation of how it meets the two conditions.

Suppose X and Y are {0, 1}-valued random variables with joint probability mass function given in...

Suppose X and Y are {0, 1}-valued random variables with joint probability mass function given in the table below p(x, y) y 0 1 x 0 0.3 0.2 1 0.1 0.4 a. Determine E(Y |X = x) and var(Y |X = x). b. Use part a) to find E(Y ), and compare with the result using the marginal distribution of Y . c. Use part a) to find var(Y ), and compare with the result using the marginal distribution of...

Let X and Y be independent discrete random variables with the following PDFs: x 0 1...

Let X and Y be independent discrete random variables with the following PDFs: x 0 1 2 f(x)=P[X=x] 0.5 0.3 0.2 y 0 1 2 g(y)= P[Y=y] 0.65 0.25 0.1 (a) Show work to find the PDF h(w) = P[W=w] = (f*g)(w) (the convolution) of W = X + Y (b) Show work to find E[X], E[Y] and E[W] (note that E[W] = E[X]+E[Y])

The joint probability distribution of variables X and Y is shown in the table below. ...............................................................................X..........................................................................

The joint probability distribution of variables X and Y is shown in the table below. ...............................................................................X....................................................................... Y 1 2 3 1 0.30 0.18 0.12 2 0.15 0.09 0.06 3 0.05 0.03 0.02 Calculate E(XY) Determine the marginal probability distributions of X and Y. Calculate E(X) and E(Y) Calculate V(X) and V(Y) Are X and Y independent? Explain. Find P(Y = 2| X = 1) Calculate COV(X,Y). Did you expect this answer? Why? Find the probability distribution...