Ch 5

Question 1

Define variance

Answer 1

Let X be rv with E[X^2]<inf.
Var(X)=E[(X-E[X])^2]

Question 2

What does Var(aX+b) equal?

Answer 2

a^2Var(X)

Question 3

What is another way of writing the variance?

Answer 3

Var(X)=E[X^2]-(E[X])^2

Question 4

What does the variance of a random variable provide a measure of?

Answer 4

Whether samples from a probability distribution are mainly clustered close to the mean or more widely spread

Question 5

Define standard deviation

Answer 5

Small sigma=sqrt(Var(X))

Question 6

What can we say about the variance of a rv if we know that it exists?

Answer 6

Var(X)>=0

Question 7

When does Var(X)=0?

Answer 7

Iff there exists c€R st P(X=c)=1. In this case c=E[X]

Question 8

What can we say about the expected value of E[XY] if X and Y are independent?

Answer 8

E[XY]=E[X]E[Y]

Question 9

What can we say about the variance of the sum of two independent rvs?

Answer 9

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

Question 10

What is variance of binomial rv?

Answer 10

Var(X)=np(1-p)

Question 11

If rv Y=aX+b where X is a continuous rv. Define pdf fY

Answer 11

fY(y)=1/|a|fX((y-b)/a)

Question 12

Monotone transformation

Answer 12

Let X be continuous rv with density fX. Define RX as set where fX is positive. Let g be a function that is différentiable and strictly monotone on RX. Then for Y=g(X)
fY(y)=fX(g^-1(y))|(g^-1)’(y)| for all y€g(RX) and equals 0 otherwise

Question 13

What is derivative of inverse?

Answer 13

(g-1)’(y)=1/g’(g^-1(y))

Question 14

What type of distribution is Y if it equals FX(X)

Answer 14

Uniform (0,1)