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 1. List all outcomes (simple events) in S 2. Find the probability for each outcome 3. Identify the value of X for each outcome 4. Find all outcomes for which X=k, for each possible k 5. 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 ** find the mean value Example: Flip a Coin 3 Times

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

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

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  and standard deviation  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  = 70 and  = 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)?