Question

Let X be a random variable with a mean of 9 and a variance of 16. Let Y be a random variable with a mean of 10 and a variance of 25. Suppose the population correlation coefficient between random variables X and Y is -0.4.

a) Find the mean of the random variable W = 3X - 5Y.

b) Find the standard deviation of the random variable Z = X + Y

Answer #1

We will use the formula above

Random variable X has mean 16 and standard deviation 9, random
variable Y has mean 12 and standard deviation 7, the correlation
between X and Y is 0.3.
How can we find the correlation between 4X - 5Y and 3Y - 1.

A random variable X is normally distributed with a mean of 121
and a variance of 121, and a random variable Y is normally
distributed with a mean of 150 and a variance of 225. the random
variables have a correlation coefficient equal to 0.6. Find the
mean and variance of the random variable :
W= 6X + 3Y

A random variable X is normally distributed with a mean of 100
and a variance of 100, and a random variable Y is normally
distributed with a mean of 180 and a variance of 324. The random
variables have a correlation coefficient equal to −0.4. Find the
mean and variance of the random variable below.W=2X−4Y
μW=
(Type an integer or a decimal.)
σ2W=
(Type an integer or a decimal.)

Suppose that X is a random variable with mean 21 and standard
deviation 4 . Also suppose that Y is a random variable with mean 42
and standard deviation 8 . Find the mean of the random variable Z
for each of the following cases
(Give your answer to three decimal places.)
a) Z = 3 + 10X
b) Z = 3X − 10
c) Z = X + Y
d) Z = X − Y
e) Z = −4X...

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)}

Let ? and ? be independent random variables. Random variable ?
has mean ?? and variance ?^2?, and random variable ? has mean ??
and variance ?^2?
a) Prove that ?[?⋅?]=??⋅??
Guidance: Start with ?[?⋅?]=ΣΣ??⋅???(?,?)??, and then use the
definition of independent random variables.
b) Use a) to prove that ???(??+??)=?^2???(?)+?^2???(?).
Guidance: Use the formula proved in the class
???(?)=?[?^2]−?^2[?].
c) Let ? =5?+3?. Find the mean and variance of ? in terms of the
means and variances of ?...

i) A random variable X has a binomial distribution with mean 6
and variance 3.6: Find P(X = 4).
ii) Let X equal the larger outcome when a pair of four-sided
dice is rolled. The pmf of X is
f(x) = (2x - 1/ 16) ; x = 1; 2; 3; 4.
Find the mean, variance and standard deviation of X.
iii) Let μ and σ^2 denote the mean and variance of the random
variable able X. Determine E [(X...

Let the random variables X and Y have joint cdf as follows:
F(x,y) = {1-e^-4x-e^-5y+e^-4x-5y, x>0, y>0
0, otherwise
Find the correlation coefficient of X and Y

The expected values, variances and standard Deviatiations for
two random variables X and Y are given in the following table
Variable
expected value
variance
standard deviation
X
20
9
3
Y
35
25
5
Find the expected value and standard deviation of the following
combinations of the variable X and Y. Round to nearest whole
number.
E(X+10) = ,
StDev(X+10) =
E(2X) = ,
StDev(2X) =
E(3X-2) = ,
StDev(3X-2) =
E(3X +4Y) = ,
StDev(3X+4Y) =
E(X-2Y) = ,
StDev(X-2Y) =

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.

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