12. The random variables X and Y have the joint distribution given below. The un

Question

12. The random variables X and Y have the joint distribution given below. The units of X arc inches and the units of Y arc centimcters. -1 -0.5 0.5 -2 1/8 -1 1/4 0 1/2 1/8 (a) Write the marginal distributions for X and Y in the table. (b) Are X and Y independent? Why or why not? Fully justify your answer using arguments from class (c) What is the conditional distribution fxjy) (Remember to show how you got your answer.) 13. Suppose W is a normally distributed random variable with mean 3 and variance 16 Find a such that Fw(a) 0.9985.

Explanation / Answer

Question 12:

a) The marginal distribution for X here is computed by adding the columns as:

P(X = -1) = 0.125
P(X = - 0.5) = 0.25
P(X = 0.5) = 0.5
P(X = 1) = 0.125

The marginal distribution for Y here is computed by adding the rows as:

P(Y = -2) = 0.125,
P(Y = -1) = 0.25,
P(Y = 1) = 0.5,
P(Y = 2) = 0.125

b) Here, we have:

P(X = -1, Y = -2) = 0.125

P(X = -1)P(Y = -2) = 0.125*0.125 which is not equal to P(X = -1, Y = -2)

Therefore X and Y are not independent here.

c) Given Y= 1, we see that X could onlt take the value of 0.5. Therefore we get the conditional PDF given Y = 1 as:

P(X = 0.5 | Y = 1) = 1

P(X = -1 | Y = 1) = P(X = -0.5 | Y = 1) = P(X = 1 | Y = 1) = 0

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