Outline of a formula sheet for STOR 435 - Introduction to Probability at UNC.

```Expected value

Expected value of geometric distribution is EX= q/p
(220)

Expect value of abs. continuous random var. EX= integral -inf to inf of xf(x)dx.
(220)

Expected value of exponential random variable, that is, lambda*e^(-lambda*x) is 1/lambda.
(222)

Algebra of expected values

Say EY=Eh(X), then EY=integral 0 to inf of h(x)f(x)dx, where f(x) is density of X. or sum 	h(x)p(x) for discrete.
(243)

NOTE: E(X^2) != (EX)^2
(244)

If X is standard normal, then EX^2 = 1
(247)

X be number of successes in n Bernoulli trials, then EX= np where n is number of trials.
(250)

E(X+Y)^2 = E(X^2+2XY+Y^2)
(253)

VX=EX^2 - (EX)^2
(262)

Variance of exponential random = 1/lamba^2
(263)

V(aX+b) = a^2 VX (linear function, variance)
(264)

Moments: u_k = EX^k. So kth moment would be integral of x^k f(x) dx.
(266,268).

[E(X+Y)]^2 = (EX)^2 + (EY)^2 + 2(EX)(EY)
(271)

Covariance: Cov(X,Y) = E(XY) - EX * EY
(272)

Note for E(XY) where there is a joint density:

```