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Question 3. The joint PMF of X and Y is f(x,y) = { 1/6, 0 ? x + y ? 2 and x, y are nonnegative integers 0, else (1) Find marginals fX and fY. (2) Are X, Y independent? (3) Find the correlation coefficient ?X,Y. (4) Compute E[X + Y]. (5) Compute E[XY^2].

          Question 3. The joint PMF of X and Y is
f(x,y) = { 1/6,    0 ? x + y ? 2 and x, y are nonnegative integers
          0,      else

(1) Find marginals fX and fY. (2) Are X, Y independent? (3) Find the correlation coefficient ?X,Y.
(4) Compute E[X + Y]. (5) Compute E[XY^2].

Show more…

Question 3. The joint PMF of X and Y is
f(x,y) = 1/6, 0 ? x + y ? 2 and x, y are nonnegative integers
0, else

(1) Find marginals fX and fY. (2) Are X, Y independent? (3) Find the correlation coefficient ?X,Y.
(4) Compute E[X + Y]. (5) Compute E[XY^2].

Added by Cathy S.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Madhur L

07/29/2023

Step 1

Let's check: f(1,0) = 1/6 ≠ (1/2)*(1/3) = 1/6 f(1,1) = 1/6 ≠ (1/2)*(1/3) = 1/6 f(1,2) = 1/6 ≠ (1/2)*(1/6) = 1/12 f(2,0) = 1/6 ≠ (1/3)*(1/3) = 1/9 f(2,1) = 1/6 ≠ (1/3)*(1/3) = 1/9 f(2,2) = 0 = (1/3)*(1/6) = 1/18 Since the joint PMF is not equal to the product of Show more…

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Question 3. The joint PMF of X and Y is given by: P(X = x, Y = y) = 1/6, if 0 < x+y < 2 and x, y are nonnegative integers 0, otherwise (1) Find the marginal PMFs fX and fY. (2) Are X and Y independent? (3) Find the correlation coefficient ρX,Y. (4) Compute E[X + Y]. (5) Compute E[XY^2].

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Transcript

00:01 To find out the marginal pms of x and y.

00:05 So for x, f of x of x equal to summation f of x comma y for all y.

00:16 So it is equal to f of 0 comma 0 plus f of 0 comma 1 plus f of 0 comma 2.

00:29 It is equal to 1 by 6 plus 1 by 6 plus 1 by 6 equal to 3 by 6 equal to 1 by 2.

00:35 And for f x of 1 equal to f x of 1 equal to f of 1 comma 0 plus f of 1 comma 1 equal to 2 by 6 equal to 1 by 2.

00:52 And f x of 2 equal to f of 2 comma 0 equal to 1 by 6.

00:56 So for y, f y of y is equal to summation f of x comma y for all x.

01:07 So here f y of 0 equal to f y of 0 comma here it is f x of 0.

01:16 F of 0 comma 0 plus f of 1 comma 0 plus f of 2 comma 0 equal to 3 by 6 equal to 1 by 2.

01:27 And f y of 1 equal to f of 0 comma 1 plus f of 1 comma 1 equal to 1 by 6 plus 1 by 6 equal to 2 by 6 1 by 3.

01:37 And f y of 2 equal to f of 0 comma 2 equal to 1 by 6 1 by 6.

01:46 So we have to find the expected values here.

01:50 So these are the marginal pms of x and y.

01:57 And in second question we have to find out they are independent or not.

02:01 So here f x of x into f y of y is equal to f x y of x y.

02:20 Then it is independent.

02:21 Here it is possible.

02:28 So therefore x and y are independent...

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