Expectation, Independence And Variance

Question 1

What is expectation of a discrete RV

Answer 1

Expectation of a discrete RV is:

E(Y)= sum(yiP(X=yi) as long as sum with mod(x)< infinity

Question 2

What is expectation of continuous RV

Answer 2

Expectation of continuous RV is:
E(X)= integral infinity down to -infinity xfx(x) dx
As long as integral with mod(x) < infinity

Question 3

What does the law of unconscious statistician state

Answer 3

Law of unconscious statistician states that:
For a continuous RV X and function g R to R
E[g(X)] = integral infinity down to - infinity g(x)fx(x) dx
Where the integral exists

Question 4

What are the properties of E(X)

Answer 4

Properties of E(X) are:
E(aX+b)= aE(X) +b
E[g(X)+ h(X)] = E[g(X)] + E[h(X)]

Question 5

What is Var(X)

Answer 5

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

Question 6

What is Var(a+bX)

Answer 6

Var(a+bX)= b^2Var(X)

Question 7

What is the joint CDF of RVs X and Y

Answer 7

Joint CDFs of RVs X and Y is:

Fx,y(x,y)= P(X<=x and Y<=y)

Question 8

When are RVs X and Y independent

Answer 8

RVs X and Y are independent if:
P(X<=x,Y<=y)=P(X<=x)P(Y<=y) for all x,y
F(x,y) = Fx Fy for all x,y

Question 9

When is a list of RVs IID

Answer 9

A list of RVs is IID when:
P(X1<=x1,…Xn<=Xn) = P(X1<=x1) …P(Xn<=xn) (independence)
And RVs X1,..,Xn are from the same distribution.
x1,…xn taken by RVs form a random sample from distribution

Question 10

What does the theorem of transformed RVs state

Answer 10

Theorem of transformed RVs states that:
Suppose X and Y are independent RVs and g and h are maps from R to R
G(X) and h(Y) are also independent