Question

Suppose X_{1}, X_{2}, X_{3}, and
X_{4} are independent and identically distributed random
variables with mean 10 and variance 16. in addition, Suppose that
Y_{1}, Y_{2}, Y_{3}, Y_{4}, and
Y_{5}are independent and identically distributed random
variables with mean 15 and variance 25. Suppose further that
X_{1}, X_{2}, X_{3}, and X_{4} and
Y_{1}, Y_{2}, Y_{3}, Y_{4}, and
Y_{5}are independent. Find Cov[bar{X} + bar{Y} + 10,
2bar{X} - bar{Y}], where bar{X} is the sample mean of
X_{1}, X_{2}, X_{3}, and X_{4} and
bar{Y} is the sample mean of Y_{1}, Y_{2},
Y_{3}, Y_{4}, and Y_{5}.

Answer #1

Suppose that X1, X2, . . . , Xn are independent identically
distributed random
variables with variance σ2. Let Y1 = X2 +X3 , Y2 = X1 +X3 and
Y3 = X1 + X2. Find the following : (in terms of σ2)
(a) Var(Y1)
(b) cov(Y1 , Y2 )
(c) cov(X1 , Y1 )
(d) Var[(Y1 + Y2 + Y3)/2]

Let
x1, x2 x3 ....be a sequence of independent and identically
distributed random variables, each having finite mean E[xi] and
variance Var（xi）.
a）calculate the var （x1+x2）
b）calculate the var（E[xi]）
c） if n-> infinite, what is Var（E[xi]）？

You are given that X1 and X2 are two independent and identically
distributed random variables with a Poisson distribution with mean
2. Let Y = max{X1, X2}. Find P(Y = 1).

Suppose that X1,X2 and X3 are independent random variables with
common mean E(Xi) = μ and variance Var(Xi) = σ2. Let V= X2−X3 and W
= X1− 2X2 + X3.
(a) Find E(V) and E(W).
(b) Find Var(V) and Var(W).
(c) Find Cov(V,W).
(d) Find the correlation coefficient ρ(V,W). Are V and W
independent?

Let X1, X2, X3, and X4 be mutually independent random variables
from the same distribution. Let
S = X1 + X2 + X3 + X4. Suppose we know that S is a Chi-Square
random variable with 2 degrees of freedom. What
is the distribution of each of the Xi?

Continuous random variables X1 and X2 with joint density
fX,Y(x,y) are independent and identically distributed with expected
value μ.
Prove that E[X1+X2] = E[X1] +E[X2].

Suppose x1, x2, x3, x4 is linearly independent in V . Prove that
x1 − x2, x2 − x3, x3 − x4, x4 is also linearly independent in V

f X1,X2,X3,X,X5 are independent and identically distributed
geometrically distributed ran
dom variables with the parameter p, compute (a) Find c.d.f. of
Ymax = max{X1,...,X5};
(b) Find p.d.f. of Ymin = min{X1,...,X5}

Let X and Y be independent and identically
distributed random variables with mean μ and variance
σ2. Find the following:
a) E[(X + 2)2]
b) Var(3X + 4)
c) E[(X - Y)2]
d) Cov{(X + Y), (X - Y)}

1.suppose that Y1 and Y2 are independent random variables
2.suppose that Y1 and Y2each have a mean of A and a variance of
B
3.suppose X1 and X2 are related to Y1 and Y2 in the following
way:
X1=C/D x Y1
X2= CY1+CY2
4.suppose A, B, C, and D are constants
What is the expected value of the expected value of X1 given
X2{E [E (X1 | X2)]}?
What is the expected value of the expected value of X2 given...

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