Question

1.) Consider two independent discrete random variables X and Y with V(X)=2 and V(Y)=5. Find V(4X-8Y-9).

2.) Consider two independent discrete random variables X and Y with SD(X)=16 and SD(Y)=9. Find SD(5X-2Y-13). (Round your answer to 1 place after the decimal point).

3.)Consider two discrete random variables X and Y with V(X)=81 and V(Y)=36 and correlation ρ=0.7. Find V(X-Y). (Round your answer to 1 digit after the decimal point).

Answer #1

1)

Var ( 4 X - 8Y - 9) = 4^{2} Var(X) + 8^{2}
var(Y) - 8 * 4 * 2 Cov (X, Y)

= 16 * 2 + 64 * 5 + 0 (For independent variables , Cov ( X , Y) = 0 )

= **352**

2)

Var ( 5X - 2Y - 13) = 5^{2} Var(X) + 2^{2}
var(Y) - 5 * 2 * 2 Cov (X, Y)

= 25 * 16^{2} + 4 * 9^{2} + 0 (For independent
variables , Cov ( X , Y) = 0 )

= 6724

SD ( 5X - 2Y - 13) = sqrt { Var { 5X - 2Y - 13 } }

= sqrt { 6724)

= **82**

3)

= Cov (X , Y) / sqrt ( Var (X) * Var(Y) )

Cov (X , Y) = * sqrt ( Var (X) * Var(Y) )

= 0.7 * sqrt ( 81 * 36)

= 37.8

Var ( X - Y) = Var(X) + Var (Y) - 2 Cov (X , Y)

= 81 + 36 - 2 * 37.8

= **41.4**

Given independent random variables with means and standard
deviations as shown, find the mean and standard deviation of each
of these variables:
Mean
SD
a) 4X
b) 4Y+3
c) 2X+5Y
X
70
14
d) 5X−4Y
e) X1+X2
Y
10
5
a) Find the mean and standard deviation for the random variable
4X.
E(4X)=_____________
SD(4X)=_______________
(Round to two decimal places as needed.)
b) Find the mean and standard deviation for the random variable
4Y+3.
E(4Y+3)= ________________
SD(4Y+3)= _____________________
(Round to...

Consider the following bivariate distribution p(x, y) of two
discrete random variables X and Y.
Y\X
-2
-1
0
1
2
0
0.01
0.02
0.03
0.10
0.10
1
0.05
0.10
0.05
0.07
0.20
2
0.10
0.05
0.03
0.05
0.04
a) Compute the marginal distributions p(x) and p(y)
b) The conditional distributions P(X = x | Y = 1)
c) Are these random variables independent?
d) Find E[XY]
e) Find Cov(X, Y) and Corr(X, Y)

Topic: Linear Combination Of Random Variables
Suppose X and Y are independent random variables with X ∼ N(1,
9) and Y ∼ N(2, 16). Find the probability that 2Y ≥ 1; find the
probability that X − Y ≥ 0.

Let X and Y be two independent random variables with
μX =E(X)=2,σX =SD(X)=1,μY =2,σY =SD(Y)=3.
Find the mean and variance of
(i) 3X
(ii) 6Y
(iii) X − Y

Problems 9 and 10 refer to the discrete random variables X and Y
whose joint distribution is given in the following table.
Y=-1
Y=0
Y=1
X=1
1/4
1/8
0
X=2
1/16
1/16
1/8
X=3
1/16
1/16
1/4
P9: Compute the marginal distributions of X and Y, and use these
to compute E(X), E(Y), Var(X), and Var(Y).
P10: Compute Cov(X, Y) and the correlation ρ for the random
variables X and Y. Are X and Y independent?

Let X and Y be independent discrete random variables with
pmf’s:
x
1
2
3
y
2
4
6
p(x)
0.2
0.2
0.6
p(y)
0.3
0.1
0.6
What is the probability that X + Y = 7

Consider independent random variables X and Y , such that X has
mean 2 and standard deviation 4, and Y has mean 1 and standard
deviation 9. Find the mean and standard deviation of the following
random variables. a) 3X b) Y + 6 c) X + Y d) X − Y e) X1 + X2,
where X1, X2 are independent copies of X.

Let X and Y be two independent random variables. Given the
marginal pdfs indicated below, find the cdf of Y/X. (Hint: Consider
two cases, 0 ≤ w ≤ 1 and 1.) (a) fx (x) =1, 0 ≤ x ≤ 1, and fγ
(y)=1, 0 ≤ y ≤ 1 (b) fx (x)=2x,0 ≤x ≤1, and fy(y)=2y, 0 ≤y ≤1

Let X and Y be discrete random variables, their joint pmf is
given as ?(x,y)= ?(? + ? − 2)/(B + 1) for 1 < X ≤ 4, 1 < Y ≤
4 Where B is the last digit of your registration number ( B=3) a)
Find the value of ? b) Find the marginal pmf of ? and ? c) Find
conditional pmf of ? given ? = 3

Consider the following independent discrete random
variables.
x = number of tornadoes detected in any given month for
state X
x
0
1
2
3
4
5
p(x)
0.55
0.17
0.14
0.09
0.03
0.02
y = number of tornadoes detected in any given month for
state Y
y
0
1
2
3
4
5
p(y)
0.40
0.30
0.15
0.11
0.01
0.03
?x = (0)(0.55) + (1)(0.17) +
(2)(0.14) + (3)(0.09) + (4)(0.03) + (5)(0.02) = 0.94
?y = (0)(0.40) +...

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