Question

(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) + Var(Y) + 2Cov(X,Y) (f) TRUE / FALSE Fo rZ∼N(0,1), P(Z >−1.2) ≈ 0.1151 (g) TRUE / FALSE If X is a Poisson random variable with parameter λ= 2, then E[X] = 2. (h) TRUE / FALSE If X is an exponential random variable with parameter λ= 2, then E[X] = 2.

Answer #1

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...

A Poisson random variable is a variable X that takes on the
integer values 0 , 1 , 2 , … with a probability mass function given
by p ( i ) = P { X = i } = e − λ λ i i ! for i = 0 , 1 , 2 … ,
where the parameter λ > 0 .
A)Show that ∑ i p ( i ) = 1.
B) Show that the Poisson random...

Let X be a Poisson random variable with parameter λ and Y an
independent Bernoulli random variable with parameter p. Find the
probability mass function of X + Y .

The random variable W = X – 3Y + Z + 2 where X, Y and Z are
three independent Normal random variables, with E[X]=E[Y]=E[Z]=2
and Var[X]=9,Var[Y]=1,Var[Z]=3.
The pdf of W is:
Uniform
Poisson
Binomial
Normal
None of the other pdfs.

true or false:
a)Var(X)=E(X^2)-E(X)^2) is always true
b)if A and B are dependent then P(A interesection B) -
P(A)P(B)=1
c)one of the desavantages of the average is it small sensibility at
data change
d) Pearson coefficient does not indicate the assimetria of a
empiric distribution

Suppose that X, Y, and Z are independent, with E[X]=E[Y]=E[Z]=2,
and E[X2]=E[Y2]=E[Z2]=5.
Find cov(XY,XZ).
(Enter a numerical answer.)
cov(XY,XZ)=
Let X be a standard normal random variable. Another random
variable is determined as follows. We flip a fair coin (independent
from X). In case of Heads, we let Y=X. In case of Tails, we let
Y=−X.
Is Y normal? Justify your answer.
yes
no
not enough information to determine
Compute Cov(X,Y).
Cov(X,Y)=
Are X and Y independent?
yes
no
not...

Let X and Y be jointly distributed random variables with means,
E(X) = 1, E(Y) = 0, variances, Var(X) = 4, Var(Y ) = 5, and
covariance, Cov(X, Y ) = 2.
Let U = 3X-Y +2 and W = 2X + Y . Obtain the following
expectations:
A.) Var(U):
B.) Var(W):
C. Cov(U,W):
ans should be 29, 29, 21 but I need help showing how to
solve.

Consider two random variables X and Y such that E(X)=E(Y)=120,
Var(X)=14, Var(Y)=11, Cov(X,Y)=0.
Compute an upper bound to
P(|X−Y|>16)

Let X denote a random variable that follows a binomial
distribution with parameters n=5, p=0.3, and Y denote a random
variable that has a Poisson distribution with parameter λ = 6.
Additionally, assume that X and Y are independent random
variables.
Derive the joint probability distribution function for X and Y.
Make sure to explain your steps.

True or false? and explain please.
(i) Let Z = 2X + 3. Then, Cov(Z,Y ) = 2Cov(X,Y ).
(j) If X,Y are uncorrelated, and Z = −5−Y + X , then V ar(Z) = V
ar(X) + V ar(Y ).
(k) Let X,Y be uncorrelated random variables (Cov(X,Y ) = 0) with
variance 1 each. Let A = 3X + Y . Then, V ar(A) = 4.
(l) If X and Y are uncorrelated (meaning that Cov[X,Y ] =...

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