Answer:
(a): Marginal pmf of x


(b): Marginal pmf of y


(c): Mean and Variance of x


(d): Mean and Variance of y


(e): The covariance and the coefficient of correlation

Step-by-step explanation:
Given
<em>x = bottles</em>
<em>y = carton</em>
<em>See attachment for complete question</em>
<em />
Solving (a): Marginal pmf of x
This is calculated as:

So:






Solving (b): Marginal pmf of y
This is calculated as:

So:






Solving (c): Mean and Variance of x
Mean is calculated as:

So, we have:




Variance is calculated as:

Calculate 




So:




Solving (d): Mean and Variance of y
Mean is calculated as:

So, we have:




Variance is calculated as:

Calculate 




So:




Solving (e): The covariance and the coefficient of correlation
Covariance is calculated as:

Calculate E(xy)

This gives:




So:




The coefficient of correlation is then calculated as:





--- approximated