Expected Value, Variance and Moment Generating Functions

Question 1

Expected Value

Answer 1

Let X be a random function, then we define its expected value E(X) = SUM_all x xP[X=x]

=SUM_all x xpX(x)

Also called mean and denoted by mu

Question 2

Some properties of E(X)

Answer 2

• E(cX) = cE(X)
• E(c) = c
• E(X + Y) = E(X) + E(Y)

Question 3

Do example on page 41

Question 4

Variance

Answer 4

Let X be a random variable. X has a variance Var(X) or sigma^2

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

=SUM_all x (x- mu)^2P[X=x]

or

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

Question 5

Some properties of Var(X)

Answer 5

• Var(cX) = c^2Var(X)
• If X is discrete:
  
  • Var(X) = E(X^2) - (E(X))^2

Question 6

Do example p.42

Question 7

K-th moment of a Random Variable

Answer 7

We call E(X^k) the K-th moment of a random variable X.

We say that the K-th moment exists if E(|X|^k) < infty

Question 8

Discrete uniform distribution

Answer 8

Question 9

Bernoulli distribution

Answer 9

Question 10

Binomial

Answer 10

E(X) = np

and

Var(X) = np(1-p)

Question 11

Poisson

Answer 11

E(X) = lambda

and

Var(X) = lambda

Question 12

Moment Generating Function

Answer 12

Let X be a discrete random variable. Then, we call

MX(t) = E(e^{tx}) = SUM_x e^{tx} P(X=x)

the moment generating function.

It is defined for all real values of t such that

SUM_x e^{tx} P(X=x)

converges.