Question

6. (6%) Consider this question related to Q3, Q4, and Q5 . Is this statement true: there exist two independent random variables X and Y such that Var [X] = Var [Y] = 1;E[X] = E[Y] = 0, and also Cov [X, Y] = 0? If true, find such example, otherwise prove why this is impossible.

7. (9%) Is it possible that Covariance [X, Y] equals to (a) 0.5 (6%) (b) 5 (3%) or some random variables X and Y. Explain! Note in this question, we do not require X and Y to be independent.

Answer #1

(a) TRUE / FALSE If X is a random variable, then (E[X])^2 ≤
E[X^2]. (b) TRUE / FALSE If Cov(X,Y) = 0, then X and Y are
independent. (c) TRUE / FALSE If P(A) = 0.5 and P(B) = 0.5, then
P(AB) = 0.25. (d) TRUE / FALSE There exist events A,B with P(A)not
equal to 0 and P(B)not equal to 0 for which A and B are both
independent and mutually exclusive. (e) TRUE / FALSE Var(X+Y) =
Var(X)...

When we say Prove or disprove the
following statements, “Prove” means you show the
statement is true proving the correct statement using at most 3
lines or referring to a textbook theorem.
“Disprove” means you show a statement is wrong by
giving a counterexample why that is not true).
Are the following statements true or not? Prove or disprove
these one by one. Show how the random variable X looks in each
case.
(a) E[X] < 0 for some random...

Consider the following joint distribution between random
variables X and Y:
Y=0
Y=1
Y=2
X=0
P(X=0, Y=0) = 5/20
P(X=0, Y=1) =3/20
P(X=0, Y=2) = 1/20
X=1
P(X=1, Y=0) = 3/20
P(X=1, Y=1) = 4/20
P(X=1, Y=2) = 4/20
Further, E[X] = 0.55, E[Y] = 0.85, Var[X] = 0.2475 and Var[Y] =
0.6275.
a. (6 points) Find the covariance between X and Y.
b. (6 points) Find E[X | Y = 0].
c. (6 points) Are X and Y independent?...

A joint density function is given by fX,Y (x, y) = ( kx, 0 <
x < 1, 0 < y < 1 0, otherwise.
(a) Calculate k
(b) Calculate marginal density function fX(x)
(c) Calculate marginal density function fY (y)
(d) Compute P(X < 0.5, Y < 0.1)
(e) Compute P(X < Y )
(f) Compute P(X < Y |X < 0.5)
(g) Are X and Y independent random variables? Show your
reasoning (no credit for yes/no answer).
(h)...

Suppose that the joint probability density function of the
random variables X and Y is f(x, y) = 8 >< >: x + cy^2 0 ≤
x ≤ 1, 0 ≤ y ≤ 1 0 otherwise.
(a) Sketch the region of non-zero probability density and show
that c = 3/ 2 .
(b) Find P(X + Y < 1), P(X + Y = 1) and P(X + Y > 1).
(c) Compute the marginal density function of X and Y...

TRUE or FALSE? Do not explain your answer.
(a) If A and B are any independent events, then P(A ∪ B) = P(A)
+ P(B).
(b) Every probability density function is a continuous
function.
(c) Let X ∼ N(0, 1) and Y follow exponential distribution with
parameter λ = 1. If X and Y are independent, then the m.g.f. MXY
(t) = e t 2 /2 1 1−t .
(d) If X and Y have moment generating functions MX and...

Let X and Y be continuous random variables with joint
distribution function F(x, y), and let g(X, Y ) and h(X, Y ) be
functions of X and Y . Prove the following:
(a) E[cg(X, Y )] = cE[g(X, Y )].
(b) E[g(X, Y ) + h(X, Y )] = E[g(X, Y )] + E[h(X, Y )].
(c) V ar(a + X) = V ar(X).
(d) V ar(aX) = a 2V ar(X).
(e) V ar(aX + bY ) = a...

A random variable X has probability density function f(x)
defined by f(x) = cx−6 if x > 1, and f(x) = 0, otherwise.
a. Find the constant c.
b. Calculate E(X) and Var(X).
c. Now assume Z1, Z2, Z3, Z4 are independent RVs whose
distribution is identical to that of X. Compute E[(Z1 +Z2 +Z3
+Z4)/4] and Var[(Z1 +Z2 +Z3 +Z4)/4].
d. Let Y = 1/X, using the formula to find the pdf of Y.

Question 6. Using only a random number
generator that simulates a random number between [0,1], simulate
1,000 iterations of the following random variables:
a) X ~ exp(5)
b) X such that f(x) = 3x2 , 0 ≤ x ≤ 1
For both (a) and (b), plot a histogram of your simulated random
variable. Estimate ?̅ and s2. Compare these values to E(X) and
Var(X).

The random variables X and Y have the joint PDF
FX,Y(x,y) = { 6*e^-(3x + 2y) 0 <= x, y
{ 0 otherwise
(a) Show whether X and Y are independent or not.
(b) Find the PDF of fX,Y |B(x,y) where B represents the event X
+ Y < 3
(c) Find fY | B(x) where B represents the event X + Y < 3

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