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

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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(X2) != (EX)2 
		(244) 

	If X is standard normal, then EX2 = 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(X2+2XY+Y2) 
		(253) 
		
	VX=EX2 - (EX)2 
		(262) 

	Variance of exponential random = 1/lamba2 
		(263) 

	V(aX+b) = a2 VX (linear function, variance) 
		(264) 

	Moments: u_k = EXk. So kth moment would be integral of xk 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: