Question

1. There is a random variable X. It has the probability distribution of f(x) = 0.55 - .075X, for 2 < x < 6. What is E(X)?

2. Continuing with the random variable X from question 1 (It has the probability distribution of f(x) = 0.55 - .075X, for 2 < x < 6): What is V(X)?

3. Let R and S be two independent and identically distributed random variables. E(R) = E(S) = 4. V(R) = V(S) = 3. Let T = R – S. What is V(T)?

4. Let V(X) = 49. What does the Cov(X,X) equal?

Answer #1

There is a random variable X. It has the probability
distribution of f(x) = 0.55 - 0.75X, for 2 < x < 6. What is
E(X)?
Continuing with the random variable X from question 1 (It has
the probability distribution of f(x) = 0.55 - .075X,
for 2 < x < 6): What is V(X)?

I roll one die and my friend rolls another die. What is the
probability that at least one of us gets an even number?
Let R and S be two independent and identically distributed
random variables. E(R) = E(S) = 4. V(R) = V(S) = 3. Let T = R – S.
What is V(T)?
Let V(X) = 49. What does the Cov(X,X) equal?

The probability distribution of a couple of random variables (X,
Y) is given by :
X/Y
0
1
2
-1
a
2a
a
0
0
a
a
1
3a
0
a
1) Find "a"
2) Find the marginal distribution of X and Y
3) Are variables X and Y independent?
4) Calculate V(2X+3Y) and Cov(2X,5Y)

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.

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 uniform random variable on (0,1), X, has density function f(x)
= 1, 0 < x < 1. Let Y = X1 + X2 where X1 and X2 are
independent and identically distributed uniform random variables on
(0,1).
1) By considering the cumulant generating function of Y ,
determine the first three cumulants of Y .

Let X be a random variable with the following probability
distribution:
Value x of X P(X=x)
4 0.05
5 0.30
6 0.55
7 0.10
Find the expectation E (X) and variance Var (X) of X. (If
necessary, consult a list of formulas.)
E (x) = ?
Var (X) = ?

A random variable x has the following probability distribution.
Determine the standard deviation of x.
x
f(x)
0
0.05
1
0.1
2
0.3
3
0.2
4
0.35
A random variable x has the following probability distribution.
Determine the expected value of x.
x
f(x)
0
0.11
1
0.04
2
0.3
3
0.2
4
0.35
QUESTION 2
A random variable x has the following probability distribution.
Determine the variance of x.
x
f(x)
0
0.02
1
0.13
2
0.3
3
0.2...

Let X be a random variable with the following probability
distribution: Value x of X P=Xx 1 0.15 2 0.55 3 0.05 4 0.15 5 0.10
Find the expectation EX and variance Var X of X .

Let random variables X and Y follow a bivariate Gaussian
distribution, where X and Y are independent and Cov(X,Y) = 0.
Show that Y|X ~ Normal(E[Y|X], V[Y|X]). What are E[Y|X] and
V[Y|X]?

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