Keep Life Simple! We live and work and dream, Each has his little scheme,

1. Keep Life Simple! We live and work and dream, Each has his little scheme, Sometimes we laugh; sometimes we cry, And thus the days go by.

2. Random Variables Types of random variables Expected values Binomial and Normal distributions

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. Types of Random Variables • A continuous random variable can take any value in one or more intervals. ** give examples • A discrete random variable can take one of a countable list of distinct values. ** give examples

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

6. How to Find the Function pdf • List all outcomes (simple events) in S • Find the probability for each outcome • Identify the value of X for each outcome • Find all outcomes for which X=k, for each possible k • P(X=k) = the sum of the probabilities for all outcomes for which X=k

7. Example: Flip a Coin 3 Times ** find pdf ** draw a plot of pdf

8. CDF of a R.V. • The cumulative distribution function (cdf) of X is: P(X<k)= sum of P(X=h) over h<k

9. Example: Flip a Coin 3 Times ** find cdf ** draw a plot of cdf

10. Important Features of a Distribution • Overall pattern • Center – mean • Spread – variance or standard deviation

11. Expected Value (Mean) • The expected value of X is the mean (average) value from an infinite # of observations of X

12. Finding Expected Value • X = a discrete r.v. • { x1, x2, …} = all possible X values • pi is the probability X = xi where i = 1, 2, … • The expected value of X is:

13. Example: Flip a Coin 3 Times ** find the mean value

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

15. Example: Flip a Coin 3 Times ** find the variance and sd

16. Binomial Random Variables • Binomial experiments: Repeat the same trial of two possible outcomes (success or failure) n times independently • The # of successes out of the n trials is called a binomial random variable

17. Examples: • Flip a fair coin 3 times (or flip 3 fair coins) • The # of defective memory chips of 50 chips • An experimental treatment for bird flu • Others?

18. 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 k = 0, 1, 2, …,n, P(X=k) =

19. Example: Pass or Fail Suppose that for some reason, you are not prepared at all for the today’s quiz. (The quiz is made of 5 multiple-choice questions; each has 4 choices and counts 20 points.) You are therefore forced to answer these questions by guessing. What is the probability that you will pass the quiz (at least 60)?

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

21. 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].

22. Normal Distribution • The “model” distribution of a continuous r.v. • The r.v. with a normal distribution is called a normal r.v. • The pdf of a normal r.v. looks like:

23. CDF of a Normal R.V. • X: a normal r.v. with mean m and standard deviation s F(a) = P(X < a) = P( ) = see Table A.1 z score

24. Example • Suppose that the final scores of ST1000 students follow a normal distribution with m = 70 and s = 10. What is the probability that a ST1000 student has final score 85 or above (grade A)? • Between 75 and 85 (grade B)? • Below 50 (F)?