1. 1 Ch5. Probability Densities Dr. Deshi Ye yedeshi@zju.edu.cn

2. 2 Outline  Continuous Random variables  Kinds of Probability distribution Normal distr. Uniform distr. Log-Normal dist. Gamma distr. Beta distr. Weibull distr.  Joint distribution  Checking data if it is normal? Transform observation to near normal  Simulation

3. 3 5.1 Continuous Random Variables  Continuous sample space: the speed of car, the amount of alcohol in a person’s blood  Consider the probability that if an accident occurs on a freeway whose length is 200 miles.  Question: how to assign probabilities?

4. 4 Assign Prob.  Suppose we are interested in the prob. that a given random variable will take on a value on the interval [a, b]  We divide [a, b] into n equal subintervals of width ∆x, b – a = n ∆x, containing the points x1, x2,..., xn, respectively.  Then Frequency

5. 5  If f is an integrable function for all values of the random variable, letting ∆x-> 0, then

6. 6 Continuous Probability Density Function  1.Shows All Values, x, & Frequencies, f(x) f(X) Is Not Probability  2.Properties (Area Under Curve) Value (Value, Frequency) Frequency f(x) ab x fxdx fx () () All X a x b   1 0,

7. 7 Continuous Random Variable Probability Probability Is Area Under Curve! f(x) X ab

8. 8 Distribution function F  Distribution function F (cumulative distribution ) Or Integral calculus :

9. 9 EX  If a random variable has the probability density find the probabilities that it will take on a value A) between 1 and 3 B) greater than 0.5

10. 10 Solution B) A)

11. 11 Mean and Variance Mean: Variance:

12. 12 K-th moment  About the original  About the mean

13. 13 Useful cheat

14. 14 Continuous Probability Distribution Models Continuous Probability Distribution UniformNormalExponentialOthers

15. 15 Normal Distribution

16. 16 5.2 The Normal Distribution  Normal probability density (normal distribution) The mean and variance of normal distribution is exactly

17. 17 The Normal Distribution Mean Median Mode  1.‘Bell-Shaped’ & Symmetrical  2.Mean, median, mode are equal  3. Random variable has infinite range

18. 18 The Normal Distribution  f(x)=Frequency of random variable x  =Population standard deviation  =3.14159; e = 2.71828  x =value of random variable (- < x < )  =Population mean

19. 19 Effect of varying parameters ( & )

20. 20 Standard normal distribution function  Standard normal distribution, with mean 0 and variance 1. Hence Normal table

21. 21 Standardize the Normal Distribution One table! Normal Distribution Standardized Normal Distribution

22. 22 Not standard normal distribution  Let, then the random Variable Z, F(z) has a standard normal distribution. We call it z-scores.  When X has normal distribution with mean and standard deviation

23. 23 Find z values for the known probability  Given probability relating to standard normal distribution, find the corresponding value z.  F(z) is known, what is the value of z?  Let be such that probability is where

24. 24 Finding Z Values for Known Probabilities.1217.1217.01 0.3.1217 Standardized Normal Probability Table (Portion) What is Z given P(Z) =.1217? Shaded area exaggerated

25. 25 Find the following values (check it in Table)

26. 26 5.3 The Normal Approximation to the binomial distribution  Theorem 5.1. If X is a random variable having the binomial distribution with parameter n and p, the limiting form of the distribution function of the standardized random variable  as n approaches infinity, is given by the standard normal distribution

27. 27 EX  If 20% of the memory chips made in a certain plant are defective, what are the probabilities that in a lot of 100 random chosen for inspection?  A) at most 15.5 will be defective  B) exactly 15 will be defective  Hint: calculate it in binomial dist. And normal distribution.

28. 28 A good rule  A good rule for normal approximation to the binomial distribution is that both np and n(1-p) is at least 15