Chapter 7: Random Variables “Horse sense is what keeps horses from betting on what people do.” Damon Runyon

7.1 Discrete and Continuous Random Variables (pp ) When the outcomes of an event that produces random results are numerical, the numbers obtained are called random variables. The sample space for the event is just a list containing all possible values of the random variable. Section 7.1 introduces the concept of a random variable and the probabilities associated with the various values of the variable.

7.1 - Continued Random variable: the outcome of a random phenomenon Discrete random variables: Have a countable number of possible values Example: Flip a coin 4 times Number of heads obtained: 0, 1, 2, 3, 4 Number of heads possible is a discrete random variable, x TTHH THHT THHHTHTHTTTH HTHHHTHTTTHT HHTHHTTHTHTT HHHHHHHTHHTTHTTTTTTT x43210 Prob(x)1/164/166/164/161/16

7.1 - Continued Continuous random variable Takes all values in an interval of numbers Has a density curve associated with it Example: x is a random number in the interval and is therefore a continuous random variable RAND function generates values of x in the interval Random numbers generated on TI83+ are rounded to 10 decimal places (so you are really looking at discrete!) Distinction between > and can be ignored.

7.1 - Continued Very common types of continuous random variables are represented in normal probability distributions Random observations from a normal distribution can be distributed with a TI83+ randNorm generates 100 random numbers from a normal distribution with and stores them into List1 SortA(L1) sorts list of random numbers in ascending order

7.2 – Means and Variances of Random Variables (pp ) If x is a discrete random variable with possible values having probabilities then

Example: A random variable x assumes the values 1, 2, 3 with respective probabilities 60%, 30%, and 10% L1L2L3L4L5L SUMS1

Law of Large Numbers The actual mean of many trials gets close to the distribution mean as more trials are made. Example: A coin is flipped numerous times Expectation: 50% of the time you’ll get a head 10 flips --many times– HIGHLY LIKELY that in some of the trials you will have 30% or less HEADS 100 flips –many times-- HIGHLY UNLIKELY that any trials will yield a percentage of HEADS that is 30% or less Try on the calculator: binomcdf(10, 0.5, 3) and binomcdf(100, 0.5, 30) These give the probabilities of flipping 10 coins with 3 or less heads and flipping 100 coins with 30 of less heads

Rule #1 for Means

Rule #2 for Means

Rule #1 for Variances

Rule #2 for Variances

Rule #2 for Variances continued

American Roulette 18 Black Numbers 18 Red Numbers 2 Green Numbers Betting $1.00 on one number has a probability of 1/38 of winning $ The probability you will lose your dollar is 37/38. Your expectation is ($35)(1/38) – ($1)(37/38) = -$ The casino takes in $ for every $1 that is wagered on the game!!