MATH 4030 – 4B CONTINUOUS RANDOM VARIABLES Density Function PDF and CDF Mean and Variance Uniform Distribution Normal Distribution

Difficulties: Probability at a value Addition rule Axiom of probability Zero probability event vs. empty event Probability density function (pdf) f(x) Area for probability  cumulative distribution function (cdf) F(x) Integral for area, fundamental theorem of calculus Approach (Sec. 5.1):

Properties of f(x) and F(x): Mean and Variance:

Uniform U( ,  ) (Sec. 5.5): Value of C: Axiom of probability Graphs of f(x) and F(x) Probability Mean E[X] and variance Var[X]

Normal N( ,  2 ) (Sec. 5.2): Standard Normal Distribution Use of Table 3 (Pages ): Given z value(s) and find the probabilities; Given probability and find the cut-off z-value(s);

7 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 .

8 From General Normal to Standard Normal If X ~ N(µ,  2 ), then the transformation Z = (X - µ) /  results Z ~ N(0, 1).

Normal Approximation to Binomial (Sec. 5.3) Or if X has binomial distribution with parameters n and p, then For large n, np>15, and n(1-p)>15. Correction for continuity