Question

1. Consider a random variable which is U(-2,5). Show work and find:

a. E(X)

b. Var(X)

c. P(1<X<3)

d. F(x) evaluated at x=4

e. f(x)

f. E(absolute value of X)

Answer #1

**Please check attachment****In last page i have given alternative(easy) to calculate (a) and (b)**

Suppose that X is a normal random variable and E(X) = −3. Find
Var(X) if P(−7 < X < 1) = 0.7888.

Let X be a binomial random variable with E(X) = 7 and Var(X) =
2.1.
(a) [5 pts] Find the parameters n and p for the binomial
distribution.
n =
p =
(b) [5 pts] Find P(X = 4). (Round your answer to four decimal
places.)
(c) [5 pts] Find P(X > 12)

If X, Y are random variables with E(X) = 2, Var(X) = 3, E(Y) =
1, Var(Y) =2, ρX,Y = −0.5
(a) For Z = 3X − 1 find µZ, σZ.
(b) For T = 2X + Y find µT , σT
(c) U = X^3 find approximate values of µU , σU

(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)...

Find E(X),Var(X),σ(X) if a random variable x is given by its
density function f(x), such that f(x)=0, if x≤0 f(x)=2x5, if
0<x≤1 f(x)=0, if x>1

Consider two random variable X and Y with joint PMF given in the
table below.
Y = 2
Y = 4
Y = 5
X = 1
k/3
k/6
k/6
X = 2
2k/3
k/3
k/2
X = 3
k
k/2
k/3
a) Find the value of k so that this is a valid PMF. Show your
work.
b) Re-write the table with the joint probabilities using the
value of k that you found in (a).
c) Find the marginal...

Suppose X is a discrete random variable with probability mass
function given by
p (1) = P (X = 1) = 0.2
p (2) = P (X = 2) = 0.1
p (3) = P (X = 3) = 0.4
p (4) = P (X = 4) = 0.3
a. Find E(X^2) .
b. Find Var (X).
c. Find E (cos (piX)).
d. Find E ((-1)^X)
e. Find Var ((-1)^X)

Let X be a continuous uniform (-2,5) random variable. Let W =
|X| Your goal is to find the pdf of W.
a)Begin by finding the sample space of W
b)Translate the following into a probability statement about X:
Fw(w) = P[W <= w] = ....
c) Consider different values of W the sample of W. Do you need
to break up the sample space into cases?
d)Find the cdf of W
e)Find the pdf of W

The density function of random variable X is given by f(x) = 1/4
, if 0
Find P(x>2)
Find the expected value of X, E(X).
Find variance of X, Var(X).
Let F(X) be cumulative distribution function of X. Find
F(3/2)

1) Suppose that X and Y are two random variables, which may be
dependent and Var(X)=Var(Y). Assume that 0<Var(X+Y)<∞ and
0<Var(X-Y)<∞. Which of the following statements are NOT true?
(There may be more than one correct answer)
a. E(XY) = E(X)E(Y)
b. E(X/Y) = E(X)/E(Y)
c. (X+Y) and (X-Y) are correlated
d. (X+Y) and (X-Y) are not correlated.
2) S.D(X ± Y) is equal to, where S.D means standard
deviation
a. S.D(X) ± S.D(Y)
b. Var(X) ± Var(Y)
c. Square...

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