Please show work. X and Y are two discrete random variables having the joint probability distribution shown. a) Find the correlation between X and Y. b) Are X & Y independent?

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Please show work.

X and Y are two discrete random variables having the joint probability distribution shown.

a) Find the correlation between X and Y.

b) Are X & Y independent?

The image displays a probability table for a joint distribution, denoted as P(X, Y), where X and Y are discrete random variables.

The table is organized with X values listed on the leftmost column and Y values listed along the top row. The intersection of each row and column represents the probability of the joint occurrence of specific X and Y values. 

Here's the breakdown of the table:

- **Y Values**: 0, 5, 7, 8

- **X Values and Corresponding Probabilities**:
  - For X = 0:
    - P(X=0, Y=0) = 0.06
    - P(X=0, Y=5) = 0.07
    - P(X=0, Y=7) = 0.05
    - P(X=0, Y=8) = 0.07
  - For X = 1:
    - P(X=1, Y=0) = 0.06
    - P(X=1, Y=5) = 0.04
    - P(X=1, Y=7) = 0.08
    - P(X=1, Y=8) = 0.09
  - For X = 2:
    - P(X=2, Y=0) = 0.07
    - P(X=2, Y=5) = 0.09
    - P(X=2, Y=7) = 0.05
    - P(X=2, Y=8) = 0.05
  - For X = 3:
    - P(X=3, Y=0) = 0.05
    - P(X=3, Y=5) = 0.08
    - P(X=3, Y=7) = 0.02
    - P(X=3, Y=8) = 0.07

This table is useful for determining the probability of any combination of values for X and Y.
Transcribed Image Text:The image displays a probability table for a joint distribution, denoted as P(X, Y), where X and Y are discrete random variables. The table is organized with X values listed on the leftmost column and Y values listed along the top row. The intersection of each row and column represents the probability of the joint occurrence of specific X and Y values. Here's the breakdown of the table: - **Y Values**: 0, 5, 7, 8 - **X Values and Corresponding Probabilities**: - For X = 0: - P(X=0, Y=0) = 0.06 - P(X=0, Y=5) = 0.07 - P(X=0, Y=7) = 0.05 - P(X=0, Y=8) = 0.07 - For X = 1: - P(X=1, Y=0) = 0.06 - P(X=1, Y=5) = 0.04 - P(X=1, Y=7) = 0.08 - P(X=1, Y=8) = 0.09 - For X = 2: - P(X=2, Y=0) = 0.07 - P(X=2, Y=5) = 0.09 - P(X=2, Y=7) = 0.05 - P(X=2, Y=8) = 0.05 - For X = 3: - P(X=3, Y=0) = 0.05 - P(X=3, Y=5) = 0.08 - P(X=3, Y=7) = 0.02 - P(X=3, Y=8) = 0.07 This table is useful for determining the probability of any combination of values for X and Y.
Expert Solution
Step 1: Formula used

Cov(X,Y)=E(XY)-E(X)E(Y)

corr(X,Y)=Cov(X,Y)V(X)V(Y)

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