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Chapter 7: Random Variables

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 367-380). When the outcomes of an event that produces random results are numerical, the numbers obtained are called random variables .

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Chapter 7: Random Variables

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

  2. 7.1 Discrete and Continuous Random Variables (pp 367-380) • 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.

  3. 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 THHH THTH TTTH HTHH HTHT TTHT HHTH HTTH THTT HHHH HHHT HHTT HTTT TTTT • x 4 3 2 1 0 • Prob(x) 1/16 4/16 6/16 4/16 1/16

  4. 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.

  5. 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

  6. 7.2 – Means and Variances of Random Variables (pp 385-404) • If x is a discrete random variable with possible values having probabilities then

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

  8. 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

  9. Rule #1 for Means

  10. Rule #2 for Means

  11. Rule #1 for Variances

  12. Rule #2 for Variances

  13. Rule #2 for Variances continued

  14. 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 $35.00. The probability you will lose your dollar is 37/38. Your expectation is ($35)(1/38) – ($1)(37/38) = -$0.0526 The casino takes in $0.0526 for every $1 that is wagered on the game!!

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