Question

Suppose the joint probability distribution of X and Y is given
by the following table.

Y=>3 6 9 X

1 0.2 0.2 0

2 0.2 0 0.2

3 0 0.1 0.1

The table entries represent the probabilities. Hence the
outcome [X=1,Y=6] has probability

0.2.

a) Compute E(X), E(X2), E(Y), and E(XY). (For all answers show
your work.) b) Compute E[Y | X = 1], E[Y | X = 2], and E[Y | X =
3].

c) In this case, E[Y | X] is linear, given by E[Y | X] = β0 +
β1X where β0 and β1 are constants. Make a plot with E[Y | X] on the
vertical axis and X on the horizontal. Can you use your plot to
deduce the values of β0 and β1?

d) When E[Y | X] is linear, a formula for β1 is β1 =
Cov(X,Y)/Var(X).

And given β1, a formula for β0 is β0 = E(Y) – β1E(X).

Does applying these formulas yield the same answers that you
deduced in part c?

e) Let u = Y – (β0 + β1X). It so happens that u can take four
possible values: 1.5, -1.5, 3, and -3. Find the joint distribution
of u and X. The first row is done for you.

u=>-3 -1.5 1.5 3 X

1 0 0.2 0.2 0

2 _____ _____ _____ _____

3 _____ _____ _____ _____

Does E(u) = 0? Does Cov(X,u) = 0?

Answer #1

MARIGINAL AND JOINT DISTRIBUTIONS
The joint distribution of X and Y is as
follows.
Values of Y
1
0
P{X=x}
Values of X
1
0.1
0.2
0.3
0
0.3
0.4
0.7
P{Y=y}
0.4
0.6
1.0
a. Find the marginal distribution of X and
Y.
b. Find the conditional distribution of X given y =
1
c. Compute the conditional expectation of Y given X=1,
E{Y=y|X=1}

Suppose y is determined by the true model
y=β0+β1x+β2z+ε, and that
β2 >0 and COV(z,x) < 0. If someone were interested
in estimating β1, and did so by using OLS to estimate y
= β0 + β1x + u, would the OLS estimator of β1
be biased or not? If it is unbiased, explain why. If it is biased,
is the bias positive or negative? Why?

Consider joint Probability distribution of two random variables
X and Y given as following
f(x,y) X
2 4 6
Y 1 0.1 0.15
0.06
3 0.17 0.1
0.18
5 0.04 0.07
0.13
(a) Find expected value of g(X,Y) = XY2
(b) Find Covariance of Cov(x,y)

Consider the joint pdf
f(x, y) = 3(x^2+ y)/11
for 0 ≤ x ≤ 2, 0 ≤ y ≤ 1.
(a) Calculate E(X), E(Y ),
E(X^2), E(Y^2), E(XY
), Var(X), Var(Y ),
Cov(X, Y ).
(b) Find the best linear predictor of Y given
X.
(c) Plot the CEF and BLP as a function of X.

The joint probability distribution of two random variables X and
Y is given in the following table
X Y →
↓
0
1
2
3
f(x)
2
1/12
1/12
1/12
1/12
3
1/12
1/6
1/12
0
4
1/12
1/12
0
1/6
f(y)
a) Find the marginal density of X and the marginal density of Y.
(add them to the above table)
b) Are X and Y independent?
c) Compute the P{Y>1| X>2}
d) Compute the expected value of X.
e)...

Suppose that X and Y are two jointly continuous random variables
with joint PDF
??,(?, ?) =
??
??? 0 ≤ ? ≤ 1 ??? 0 ≤ ? ≤ √?
0
??ℎ??????
Compute and plot ??(?) and ??(?)
Are X and Y independent?
Compute and plot ??(?) and ???(?)
Compute E(X), Var(X), E(Y), Var(Y), Cov(X,Y), and
Cor.(X,Y)

Suppose you are given the following
simple dataset, regress Y on X:
y=β0+β1x+u
X
Y
1
2
2
4
6
6
Calculate β0 andβ1Show algebraic
steps.
Interpret β0 and β1
Calculate the predicted(fitted)value of each observation
Calculate the residual ofeach observation
When x=3, what is the predicted value of Y?
Calculate SSR, SST, and then SSE.
How much of the variation in Y is explained by X?
8)Calculate the variance estimator

If the joint probability distribution of X and Y is given
by:
f (x, y) = 3k (x + y), for x = 0, 1, 2, 3; y = 0, 1, 2.
a) .- Find the constant k.
b) .- Using the table of the joint distribution and the
marginal distributions, determine if variable X and variable Y are
independent.

Let the joint probability (mass) function of X and Y be given by
the following:
Values of X
1 2
1 1/3 1/6
Value of Y 2 1/6 1/3
(a) Find E(X + Y ). (b) Find E(min(X; Y )). (c) Find E(XY ).
(d) Find Cov(X; Y ) and Corr(X; Y ). Check both of them are
positive.

The random variables X and Y have a joint p.d.f. given by
f(x,y) = (3(x +y −xy))/7 for 0 ≤ x ≤ 1 and
0 ≤ y ≤ 2. Find the following.
(a) E[X], E[Y ]
(b) Cov[X,Y]

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