Question

Problems

1. Two independent random variables X and Y

have the probability distributions as follows:

X 1 2 5

P (X) 0.2 0.5 0.3

Y 2 4

P (Y) 0.7 0.3

a) Let T = X + Y. Find all possible values of T.

Compute μ and . T σ T

b) Let U = X - Y. Find all possible values of U.

Compute μ U and σ U .

c) Show that μ T

= μ X + μ Y

d) Show that μ U

= μ X − μ Y

e) Show that σ X±Y

=/ σ X ± σ Y

f) Show that σ

2

T

= σ

2

U

= σ

2

X + σ

2

Y

2) Let X be a random variable. Prove that the sum of all deviations from E(X) is always zero. Hints: use the properties of P(X) and basic algebra.

3) Let X be a discrete random variable. Prove that sigma (X) can also be written as sigma (X) = Sqrt ( sum [x^2 times P(x)] - E(x))

Answer #1

Let x and Y be two discrete random variables, where x Takes
values 3 and 4 and Y takes the values 2 and 5. Let furthermore the
following probabilities be given:
P(X=3 ∩ Y=2)= P(3,2)=0.3,
P(X=3 ∩ Y=5)= P(3,5)=0.1,
P(X=4 ∩ Y=2)= P(4,2)=0.4 and
P(X=4∩ Y=5)= P(4,5)=0.2.
Compute the correlation between X and Y.

Let x and Y be two discrete random variables, where x Takes
values 3 and 4 and Y takes the values 2 and 5. Let furthermore the
following probabilities be given:
P(X=3 ∩ Y=2)= P(3,2)=0.3,
P(X=3 ∩ Y=5)= P(3,5)=0.1,
P(X=4 ∩ Y=2)= P(4,2)=0.4 and
P(X=4∩ Y=5)= P(4,5)=0.2.
Compute the correlation between X and Y.

Let X and Y be independent discrete random variables with
pmf’s:
x
1
2
3
y
2
4
6
p(x)
0.2
0.2
0.6
p(y)
0.3
0.1
0.6
What is the probability that X + Y = 7

Let X and Y be a random variables with the joint probability
density function fX,Y (x, y) = { e −x−y , 0 < x, y < ∞ 0,
otherwise } . a. Let W = max(X, Y ) Compute the probability density
function of W. b. Let U = min(X, Y ) Compute the probability
density function of U. c. Compute the probability density function
of X + Y .

. Let X and Y be two discrete random variables. The range of X
is {0, 1, 2}, while the range of Y is {1, 2, 3}. Their joint
probability mass function P(X,Y) is given in the table below:
X\Y 1
2
3
0
0
.25 0
1
.25
0
.25
2
0
.25 0
Compute E[X], V[X], E[Y], V[Y], and Cov(X, Y).

Let X, Y be two random variables with a joint pmf
f(x,y)=(x+y)/12 x=1,2 and y=1,2
zero elsewhere
a)Are X and Y discrete or continuous random variables?
b)Construct and joint probability distribution table by writing
these probabilities in a rectangular array, recording each marginal
pmf in the "margins"
c)Determine if X and Y are Independent variables
d)Find P(X>Y)
e)Compute E(X), E(Y), E(X^2) and E(XY)
f)Compute var(X)
g) Compute cov(X,Y)

Suppose that you have two discrete random variables X and Y with
the following joint probability distribution, which is similar to
the example in class. Fill in the marginal probabilities below.
Possible Values of X
Possible
Values
of Y
1
2
3
4
1
0
18
18
14
2
18
14
18
0
Please input the exact answer in either decimal or fraction
form.

(a) Given two independent uniform random variables X, Y in the
interval (−1, 1), find E |X − Y |.
(b) Let X, Y be as in (a). Find the support and density of the
random variable Z = |X − Y |.
(c) From (b), compute the mean of Z and check whether you get
the same answer as in (a)

Suppose X,Y are discrete random variables, each taking only two
distinct values.
Prove that if E(XY)=E(X)E(Y) then X,Y are independent (Be aware
that you have to prove E(XY) =E(X)E(Y) -> X,Y independent and
NOT the converse)

Let two random variables X and Y satisfy Y |X = x ∼ Poisson (λx)
for all possible values x from X, with λ being an unknown
parameter. If (x1, Y1), ...,(xn,
Yn) is a random sample from the random variable Y |X =
x, construct the estimator for λ using the method of maximum
likelihood and determine its unbiasedness.

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