Question

Suppose X is a discrete random variable that takes on integer values between 1 and 10, with variance Var(X) = 6. Suppose that you define a new random variable Y by observing the output of X and adding 3 to that number. What is the variance of Y? Suppose then you define a new random variable Z by observing the output of X and multiplying that by -4. What is the variance of Z?

Answer #1

**Solution:**

Given: Variance of X is : Var(X) = 6

**Part a)** Define a new random variable Y by
observing the output of X and adding 3 to that number.

Y = X + 3

Then variance of Y is:

Var(Y) = Var(X+3)

Var(Y) = Var(X) +Var(3)

Var(Y) = 6 + 0

( Since variance of Constant is 0)

Thus

**Var(Y) = 6**

**Part b)** Define a new random variable Z by
observing the output of X and multiplying that by -4.

Z = (-4)X

Then find Variance of Z.

Var(Z) = Var( (-4) X )

Var(Z) = (-4)^{2} * Var(X)

(since Var( aX) = a^{2} * Var(X) )

Var(Z) = 16 * 6

**Var(Z) = 96**

Let x and Y be two discrete random variables, where x Takes
values 3 and 4 and Y takes the values 2 and 5. Let furthermore the
following probabilities be given:
P(X=3 ∩ Y=2)= P(3,2)=0.3,
P(X=3 ∩ Y=5)= P(3,5)=0.1,
P(X=4 ∩ Y=2)= P(4,2)=0.4 and
P(X=4∩ Y=5)= P(4,5)=0.2.
Compute the correlation between X and Y.

Let x and Y be two discrete random variables, where x Takes
values 3 and 4 and Y takes the values 2 and 5. Let furthermore the
following probabilities be given:
P(X=3 ∩ Y=2)= P(3,2)=0.3,
P(X=3 ∩ Y=5)= P(3,5)=0.1,
P(X=4 ∩ Y=2)= P(4,2)=0.4 and
P(X=4∩ Y=5)= P(4,5)=0.2.
Compute the correlation between X and Y.

Suppose a random variable X takes on the value of -1 or 1, each
with the probability of 1/2. Let y=X1+X2+X3+X4, where X1,....X4 are
independent. Find E(Y) and Find Var(Y)

Q6/
Let X be a discrete random variable defined by the
following probability function
x
2
3
7
9
f(x)
0.15
0.25
0.35
0.25
Give P(4≤ X < 8)
ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
Q7/
Let X be a discrete random variable defined by the following
probability function
x
2
3
7
9
f(x)
0.15
0.25
0.35
0.25
Let F(x) be the CDF of X. Give F(7.5)
ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
Q8/
Let X be a discrete random variable defined by the following
probability function :
x
2
6...

Suppose X is a discrete random variable with probability mass
function given by
p (1) = P (X = 1) = 0.2
p (2) = P (X = 2) = 0.1
p (3) = P (X = 3) = 0.4
p (4) = P (X = 4) = 0.3
a. Find E(X^2) .
b. Find Var (X).
c. Find E (cos (piX)).
d. Find E ((-1)^X)
e. Find Var ((-1)^X)

For a discrete random variable, the probability of the random
variable takes a value within a very small interval must be
A.
zero.
B.
very small.
C.
close to 1.
D.
none of the above.
QUESTION 10
The area under the density function in a certain interval of a
continuous random variable represents
A.
randomness.
B.
the area of one rectangle.
C.
the probability of the interval.
D.
none of the above.
QUESTION 11
For any random variable, X, E(X)...

let x be a discrete random variable with positive integer
outputs.
show that P(x=k) = P( x> k-1)- P( X>k) for any positive
integer k.
assume that for all k>=1 we have P(x>k)=q^k. use (a) to
show that x is a geometric random variable.

A random variable X takes values between -2 and 4 with
probability density function (pdf)
Sketch a graph of the pdf.
Construct the cumulative density function (cdf).
Using the cdf, find )
Using the pdf, find E(X)
Using the pdf, find the variance of X
Using either the pdf or the cdf, find the median of
X

A random variable X with a beta distribution takes on values
between 0 and 1, with unknown α and β.
a) Use the method of moments to obtain an estimator
forαandβ.
b) Are the estimators sufficient statistics?

Suppose that X is a discrete random variable with ?(? = 1) = ?
and ?(? = 2) = 1 − ?. Three independent observations of X are made:
(?1, ?2, ?3) = (1,2,2).
a. Estimate ? through the sample mean (this is an example of the
“method of moment” for estimating a parameter).
b. Find the likelihood function and MLE for ?.

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