econometrics 1. Suppose random variables X and Y appear together in the followin

Question

econometrics

1. Suppose random variables X and Y appear together in the following way: X 20 21 0 Y|4|-2|2|4|-11 Assume that each observation is equally likely. (a) Find the joint probability mass function, fxy (b) Find the marginal probability mass function, the distribution function, and the expectation of Y (c) Find the conditional expectation of Y given X =x, for each value of x. (d) Find the conditional expectation of Y given X. Find the expectation of the conditional expectation (i.e., E EIYIX]),and verity that it equals the unconditional expectation of Y (i.e., EIY]D).

Explanation / Answer

(a)

Since each of the given outcomes are equally likely so probability of each outcome is 1/5 =0.20.

Following table shows the joint pdf:

(B)

Following table shows the marginal pdf of X and Y :

Following table shows the marginal pdf of Y, distribution function of Y and expectation of Y:

The expected value of Y is

E(Y) = 1.4

(c)

Following table shows the conditional pdf of Y given X=x for each value of x:

---------

Now E(Y|X=0):

So,

E(Y|X=0) = -1.5

--------------

Now E(Y|X=1):

So,

E(Y|X=1) = 4

------------------------

Now E(Y|X=2):

So,

E(Y|X=2) = 3

-------------------------------------

(d)

Following table shows the calculations:

E[E(Y|X=x)] = 1.4 = E(Y)

X 0 1 2 -1 0.2 0 0 Y -2 0.2 0 0 2 0 0 0.2 4 0 0.2 0.2

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