MATH 1107 Elementary Statistics Lecture 8 Random Variables

Math 1107 – Random Variables In Class Exercise with Dice

Math 1107 – Random Variables Probability Distribution for a Dice Roll: XP(X) 11/

Math 1107 – Random Variables What is the mean? What is the standard deviation? Are these outcomes discrete or continuous? What is the probability of each outcome? What is the probability distribution? Are these outcomes dependent upon the previous roll?

Math 1107 – Random Variables There are two things that must be true for every probability distribution: 1.The Summation of all probabilities must equal 1; 2.Every individual probability must be between 0 and 1.

Math 1107 – Random Variables Examples for consideration…Are the following probability distributions? XP(X) Green.2 Blue.5 Red.1 Purple1.2 XP(X)

Math 1107 – Random Variables Important Formulas and applications: 1.μ = Σ [x*P(x)] From the (correct) Probability table for Dice: μ = (1*1/6)+(2*1/6)+(3*1/6)+(4*1/6)+(5*1/6)+(6*1/6) = 3.5

Math 1107 – Random Variables Important Formulas and applications: 2.σ 2 = Σ [(x- μ ) 2 * P(x)] From the Probability table for Dice: σ 2 = ((1-3.5) 2 *.1667)+((2-3.5) ) 2 *.1667)… +((6-3.5) ) 2 *.1667) = 2.92

Math 1107 – Random Variables Important Formulas and applications: 3.σ = SQRT(Σ [(x- μ ) 2 * P(x)]) From the Probability table for Dice: σ = SQRT( 2.92) = 1.71

Math 1107 – Random Variables An important note on rounding…keep your numbers in your calculator/computer and only round at the end!

Math 1107 – Random Variables What is an unusual event? When should we be suspect of results? Ultimately, you need to KNOW YOUR DATA to determine what makes sense or not. But here is a rule of thumb – If an event is more than 2 standard deviations away from the mean, it is “unusual”: μ + or - 2 σ

Math 1107 – Random Variables Lets say that you are a teacher. Joe and Jimmy sit next to each other in class. Here are their grades on the last 5 quizzes: QuizJoeJimmy Joe’s average = 94 Jimmy’s average (1 st 4) = 50, with a std of His score on quiz 5 is 33 std from the mean.

Math 1107 – Random Variables How do power companies detect people growing illegal plants in their homes? How do Pit Bosses determine who to throw out of a casino? How do regulatory agencies determine which athletes to test for illegal substances? The rate of autism is approximately 1 in 166. What is the probability of having a child with autism? Now, what is the probability of having 2 children with autism? What would you conclude?

Math 1107 – Random Variables Example from Page 190: Lets say that we have determined that the probability of having a girl follows the distribution on page 183. A process called “MICROSORT” enabled 13 out of 14 couples to have a girl, which was their preference. Is this process successful? According to the table on page 183, the probability of getting 13 out of 14 girls is.001. This suggests that MICROSORT’s performance is successful.

Math 1107 – Random Variables Expected Values: Knowing something about the distribution of events, enables us to create an expected value. This is calculated as: E(x) = Note that this is a similar calculation to the mean of a distribution. Σ(x* P(x))

Math 1107 – Random Variables Would you want to play a game where you had a 75% chance of winning $5 and a 25% chance of losing $10? The expected value of this game is: (.75*5)+(.25*-10) = $1.25 Assuming that you are “risk neutral” you would be willing to pay no more than $1.25 to play this game.

Math 1107 – Random Variables Would you be willing to play a lottery for $1 where the chances of winning $100K were 1/1000? The expected value of this game is: (1/1000* 100,000)+(999/1000*-1) = $99 Would you be willing to play a lottery for $1 where the chances of winning $10M were 1/15,625,000,000? (this is 6 number 1- 50) The expected value is: (1/15,625,000,000)*10,000,000)+(15,624,999,999/15,625,000, 000*-1) =

Math 1107 – Random Variables When you give a casino $5 for a bet on the number 10 in roulette, you have a 1/38 probability of winning $175 and a 37/38 probability of losing $5. What is your expected value? (175*1/38)+(-5*37/38) = -$.26 Should you play?