Question

Let X_{1} and X_{2} be two independent geometric
random variables with the probability of success 0 < p < 1.
Find the joint probability mass function of (Y_{1},
Y_{2}) with its support, where Y_{1} =
X_{1} + X_{2} and Y_{2} =
X_{2}.

Answer #1

Let X1 and X2 be independent random
variables with joint pdf f(x1, x2)
=x1e^−(x1+x2), 0< x1<∞,
0< x2<∞. Y1= 2X1 and
Y2=X2−X1.
I) Find g(y1, y2), the joint pdf of
Y1, Y2 Include and draw the support. II) Find
g1(y1), the marginal pdf of Y1.
III) Find E(Y1).

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 f(x1, x2) = 1 , 0 ≤ x1 ≤ 1 , 0 ≤ x2 ≤ 1 be the joint pdf of
X1 and X2 . Y1 = X1 + X2 and Y2 = X2 .
(a) E(Y1) .
(b) Var(Y1)
(c) Consider the marginal pdf of Y1 , g(y1) . What is value of
g(y1) where y1 = 1/3 and y1 = 6/4 ?

Suppose that X1 and X2 are independent continuous random
variables with the same probability density function as: f(x) = ( x
2 0 < x < 2, 0 otherwise. Let a new random variable be Y =
min(X1, X2,).
a) Use distribution function method to find the probability
density function of Y, fY (y).
b) Compute P(Y > 1).

1. An electronic system has two different types of components in
joint operation. Let X1 and X2 denote the
Random Length of life in hundreds of hours of the components of
Type I and Type II (Type 1 and Type 2), respectively. Suppose that
the joint probability density function (pdf) is given by
f(x1, x2) = { (1/8)y1
e^-(x1 + x2)/2, x1 > 0,
x2 > 0
0 Otherwise.
a.) Show that X1 and X2 are
independent.
b.) Find E(Y1+Y2)...

Suppose that X1 and X2 are independent continuous random
variables with the same probability density function as: f(x) = ( x
2 0 < x < 2, 0 otherwise. Let a new random variable be Y =
min(X1, X2,).
a) Use distribution function method to find the probability
density function of Y, fY (y).
b) Compute P(Y > 1).
c) Compute E(Y )

Suppose X1, X2, X3, and
X4 are independent and identically distributed random
variables with mean 10 and variance 16. in addition, Suppose that
Y1, Y2, Y3, Y4, and
Y5are independent and identically distributed random
variables with mean 15 and variance 25. Suppose further that
X1, X2, X3, and X4 and
Y1, Y2, Y3, Y4, and
Y5are independent. Find Cov[bar{X} + bar{Y} + 10,
2bar{X} - bar{Y}], where bar{X} is the sample mean of
X1, X2, X3, and X4 and
bar{Y}...

Let X1 and X2 be independent random variables such that X1 ∼ P
oisson(λ1) and X2 ∼ P oisson(λ2). Find the distribution of Y = X1 +
X2.s

Let Y be the liner combination of the independent random
variables X1 and X2 where Y = X1 -2X2
suppose X1 is normally distributed with mean 1 and standard
devation 2
also suppose the X2 is normally distributed with mean 0 also
standard devation 1
find P(Y>=1) ?

(i) Find the marginal probability distributions for the random
variables X1 and X2 with joint pdf
f(x1, x2) = 12x1x2(1-x2) , 0 < x1 <1 0 < x2 < 1 0 ,
otherwise
(ii) Calculate E(X1) and E(X2)
(iii) Are the variables X1 ¬and X2 stochastically
independent?

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