STATS 425 Lecture Notes - Lecture 7: Probability Distribution, Random Variable, Exponential Distribution
Document Summary
Cumulative distribution function: the cdf of a continuous rv is defined exactly the same as for discrete rvs, f(x) = p(x<=x) = p(x in (-infinity, x], df/dx = f(x) Expectation of continuous rvs: e(x) = xf (x) dx. Variance of continuous rvs: var(x) = e[(x-e(x))^2] Properties of expected value & variance: var(x) = e(x^2) (e(x))^2, e(ax+b) = ae(x) +b, var(ax+b) = a^2var(x) The standard normal distribution: z has the standard normal distribution if it has pdf f(x) =(1/(sqrt(2pi)))(e^(-x^2)/2) f(x) is symmetric about x = 0, so e(x) = 0, var (x) = 1. If x has density f(x) = (1/(sqrt(2 ^2)))exp{(-(x-u)^2/(2 ^2))} then x has the normal distribution: x~n(u, 2 ^2, e(x) = u, var(x) = ^2. If z~n(0, 1) then z is standard normal: x=u+ z, then x~n(u, ^2) If x~binomial (n, p) and n is large enough, then x is approximately n(np, np(1-p): this approximation is reasonably good for np(1-p)>10, p(x<=k) is usually approximated by p(y
