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

read more
Expected value

	Expected value of geometric distribution is EX= q/p 

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

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

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.  

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

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

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

	E(X+Y)^2 = E(X^2+2XY+Y^2) 
	VX=EX^2 - (EX)^2 

	Variance of exponential random = 1/lamba^2 

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

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

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

	Covariance: Cov(X,Y) = E(XY) - EX * EY 
	Note for E(XY) where there is a joint density: