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MATH 4030 – 4B CONTINUOUS RANDOM VARIABLES Density Function PDF and CDF Mean and Variance Normal Distribution Difficulties: • Probability at a value • Addition rule • Axiom of probability • Zero probability event vs. empty event Approach (Sec. 5.1): • Probability density function (pdf) f(x) • Area for probability cumulative distribution function (cdf) F(x) • Integral for area, fundamental theorem of calculus Properties of f(x) and F(x): 1) f ( x) 0 2) f ( x)dx 1. - x 3) F ( x ) P( X x) xdx f1 x(t )dt1 64 1 65 . 4 4 1 2 3 - 3 1 3 3 b 4) P(a x b) P(a x b) f ( x )dx F (b) F (a ) a Mean and Variance: xf ( x )dx - 2 2 2 ( x ) f ( x ) dx 2 - Normal N(, 2) (Sec. 5.2): 1 f ( x) e 2 Mean xf ( x )dx x 2 2 2 -4 -3 -2 -1 0 1 , x . Variance 2 2 x f ( x ) dx Standard Normal N(0, 1): 1 e 2 f ( x) x F ( x) x2 2 1 e 2 , x . t 2 2 dt , x . 2 3 4 5 z notation: A z-value that the probability for Z to be greater than this value is exactly . Or the cut point of the standard normal curve that makes the area of the right tail exactly . From General Normal to Standard Normal If X ~ N(µ, 2), then the transformation Z = (X - µ) / results Z ~ N(0, 1). 6 Normal Approximation to Binomial (Sec. 5.3) Bi (n, p ) N (np, np(1 p )) Or if X has binomial distribution with parameters n and p, then X np Z ~ N (0,1) np(1 p) • For large n, np>15, and n(1-p)>15. • Correction for continuity