1. 1 If we can reduce our desire, then all worries that bother us will disappear.

2. 2 Random Variables and Distributions Distribution of a random variable Binomial and Poisson distributions Normal distributions

3. 3 What Is a Random Variable? The numerical outcome of a random circumstance is called a random variable. Eg. Toss a dice: {1,2,3,4,5,6} Height of a student A random variable (r.v.) assigns a number to each outcome of a random circumstance. Eg. Flip two coins: the # of heads

4. 4 Types of Random Variables A continuous random variable can take any value in one or more intervals. ** eg. Height, weight, age A discrete random variable can take one of a countable list of distinct values. ** eg. # of courses currently taking

5. 5 Distribution of a Discrete R.V. X = a discrete r.v. x = a number X can take The probability distribution function (pdf) of X is: P(X = x)

6. 6 Example: Birth Order of Children ** pdf: Table 7.1 on page 163 ** histogram of pdf: Figure 7.1

7. 7 Important Features of a Distribution Overall pattern Central tendency – mean Dispersion – variance or standard deviation

8. 8 Calculating Mean Value X = a discrete r.v. { x1, x2, …} = all possible X values pi is the probability X = xi where i = 1, 2, … The mean of X is:

9. 9 Variance & Standard Deviation Notations as before Variance of X: Standard deviation (sd) of X:

10. 10 Example: Birth Order of Children

11. 11 Bernoulli and Binomial Distributions A Bernoulli trial is a trial of a random experiment that has only two possible outcomes: Success (S) and Failure (F). The notational convention is to let p = P(S). Consider a fixed number n of identical (same P(S)), independent Bernoulli trials and let X be the number of successes in the n trials. Then X is called a binomial radon variable and its distribution is called a Binomial distribution with parameters n and p. Read the handout for bernoulli and binomial distributions.

12. 12 PDF of a Binomial R.V. p = the probability of success in a trial n = the # of trials repeated independently X = the # of successes in the n trials For x = 0, 1, 2, …,n, P(X=x) =

13. 13 Mean & Variance of a Binomial R.V. Notations as before Mean is Variance is

14. Brief Minitab Instructions Minitab: Calc>> Probability Distributions>> Binomial; Click ‘probability’, ‘input constant’ and n, p, x Minitab Output: Binomial with n = 3 and p = 0.29 x P( X = x ) 2 0.179133 14

15. The Poisson Distribution a popular model for discrete events that occur rarely in time or space such as vehicle accident in a year The binomial r.v. X with tiny p and large n is approximately a Poisson r.v.; for example, X = the number of US drivers involved in a car accident in 2008 Read the Poisson distribution handout. 15

16. 16

17. Brief Minitab Instructions Minitab: Calc>> Probability Distributions>> Poisson; Click ‘probability’, ‘input constant’ and  x Minitab Output: Poisson with mean = 2.4 x P( X = x ) 1 0.217723 17

18. 18 Distribution of a Continuous R.V. The probability density function (pdf) for a continuous r.v. X is a curve such that P(a < X <b) = the area under it over the interval [a,b].

19. 19 Normal Distribution Its density curve is bell-shaped The distribution of a binomial r.v. with n= ∞ The distribution of a Poisson r.v. with l= ∞ Read the normal distribution handout.

20. 20

21. 21

22. 22 Standard Normal Distribution X: a normal r.v. with mean  and standard deviation  Then  is a normal r.v. with mean 0 and standard deviation 1; called a standard normal r.v.

23. Brief Minitab Instructions Minitab: Calc>> Probability Distributions>> Normal; Click what are needed Minitab Output: Inverse Cumulative Distribution Function Normal with mean = 0 and standard deviation = 1 P( X <= x ) x 0.95 1.64485 Cumulative Distribution Function Normal with mean = 0 and standard deviation = 1 x P( X <= x ) 1.64485 0.950000 23

24. 24 Example: Systolic Blood Pressure Let X be the systolic blood pressure. For the population of 18 to 74 year old males in US, X has a normal distribution with  = 129 mm Hg and  = 19.8 mm Hg. What is the proportion of men in the population with systolic blood pressures greater than 150 mm Hg? What is the 95-percentile of systolic blood pressure in the population?