1. Continuous Random Variables Chap. 12

2. COMP 5340/6340 Continuous Random Variables2 Preamble Continuous probability distribution are not related to specific “experiments” There are multiple “shapes” (with parameters) that fulfill the probability distribution condition (area under shape equal to one. In general, you plot your sample data and try to find a shape that fits sample data.

3. COMP 5340/6340 Continuous Random Variables3 Uniform Distribution a b

4. COMP 5340/6340 Continuous Random Variables4 Standard Normal Distribution (One of most important..)  is the mean  is the standard deviation m 

5. COMP 5340/6340 Continuous Random Variables5 Standard Normal Deviation (How to use) Standard integration techniques cannot be used Tables exist for For any other random variable X with mean  and standard deviation , we can use random variable Z such that

6. COMP 5340/6340 Continuous Random Variables6 Gamma Distributions Standard Normal distribution is symmetric… Need of skewed p.d.f… Preliminary: Gamma function  (  ) is defined: P.d.f of a gamma distribution is defined as:

7. COMP 5340/6340 Continuous Random Variables7 Gamma Distributions (2) Probability Density Function k=   = 

8. COMP 5340/6340 Continuous Random Variables8 Weibull Distribution Heavily used in fault-tolerance, reliability: lifetime, time between failures… Weibull with parameters  and 

9. COMP 5340/6340 Continuous Random Variables9 Standard Weibull Distribution ( 

10. COMP 5340/6340 Continuous Random Variables10 Measures of Central Tendency Mean (arithmetic, geometric, harmonique) Median

11. COMP 5340/6340 Continuous Random Variables11 Probability Plots Used to identify probability distribution of a sample data Building histograms with sample data is not always precise (e.g., small sample) Rather, build a probability plot based on pairs (X,Y) (n=sample size): –X= [100 (i - 0.5)/n] th percentile –Y= i th sample

12. COMP 5340/6340 Continuous Random Variables12 Normal Probability Plot If the underlying probability distribution is a normal distribution: –Plot is a straight line –With slope  (Standard deviation) –With intercept  Mean)

13. COMP 5340/6340 Continuous Random Variables13 Sum of Random Variables Given a linear combination of n random variables: –Y = a 1 X 1 +…..+ a n X n E(Y) = a 1 E(X 1 ) +…..+ a n E(X n ) V(Y) = a 1  1 2 +…..+ a n  n 2

14. COMP 5340/6340 Continuous Random Variables14 Average of Random Variables (Corollary) Let X i a random variable with mean  and standard deviation . Let a i = 1/n Y = (X 1 +…..+X n )/n E(Y) = ? V(Y) = ?

15. COMP 5340/6340 Continuous Random Variables15 Central Limit Theorem Let X 1, X 2,…,X n be a random sample from a distribution with mean  and variance  2. If n is sufficiently large, the average of these random variables has approximately a normal distribution with mean  and variance  2 /n. –The larger n, the better the approximation