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

read moreExpected 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: