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Random Variables Lesson chapter 16 part A Expected value and standard deviation for random variables

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What is a Random Event: Picking a card from a deck of cards: Tossing a coin: Etc... Can you think of any others? (pg 1.2 list 3 others ) Picking a student in a class looking for Jan birthdays: (1.4) An event with a list of outcomes where the outcome of the event is random

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What is a Random Variable: Defined as a capital X that we never solve for. A function that maps for us a random event to numerical outcomes, and the total probability of all outcomes is 1 Or a function that maps a sample space into real numbers. Picking a card from a deck of cards: (pg 1.3-1.4) Assign a value to the cards - $0.25 for face cards, $0.5 for aces, 0 for others Tossing a coin: (1.5-1.6) $2.00 for a head and $-1.00 for a tail Convert the random events you came up with to random variables? (pg 1.9 – 3 questions) Picking a student in a class looking for Jan birthdays (1.7-1.8) Jan b-day = 12 points versus non-Jan b-day = 5 points

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Now lets look at the math behind random variables – Finding the average outcome of a random variable or what you can expect to win!

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What is the average of a random variable? 9, 6, 5, 3, 3, 3, 6, 9, 2, 4 What is the average of this set of numbers? (pg 2.1) Ok, now lets do it a different way: Lets look at the formula for finding the average: 9+6+5+3+3+3+6+9+2+4 10

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9, 6, 5, 3, 3, 3, 6, 9, 2, 4 Lets do it a different way: (pg 2.2a) (2+3+3+3+4+5+6+6+9+9) /10 Use algebra: (pg 2.2b) (1x2 + 3x3 + 1x4 + 1x5 + 2x6 + 2x9) /10 Break it apart: (pg 2.3) Rewrite it: (pg 2.4) Change to percents: 10%(2) + 30%(3) + 10%(4) + 10%(5) + 20%(6) + 20%(9) This formula is called the Expected Value!!

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What is a probability model?

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If you select a face card you get $0.50, if you get any other card you get nothing. It costs $0.25 play. Is it worth playing? (what is the average pay out?) Create a Probability Model for the game. OutcomesFace CardsAll OthersTo Play Value - x P(X = x) $ 0.5$ 0.0$ -0.25 12 / 5240 / 52 You can answer 3 different questions here: 1- what is the average payout of the game? 2- what is the average winning of the game? 3- what is the average cost of the game? (pg 3.2) (pg 3.3) (pg 3.4) (pg 3.5 answer all 3 questions)

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Let’s look at the actual formulas now.

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Expected Value (mean) Over “the long haul” how much to you expect to pay/get? Can the expected value can be modeled with BoB!?! YES! Now you can find the probability of a random variable’s payout.

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Expected Value Credit unions often offer life insurance on their members. The general policy pays $1000 for a death and $500 for a disability. What is the expected value for the policy? The payout to a policyholder is the random variable and the expected value is the average payout per policy. To find E(X) create a probability model (a table with all possible outcomes and their probabilities). (pg 4.1-4.2)

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Probability Model Credit unions often offer life insurance on their members. The general policy pays $1000 for a death and $500 for a disability. What is the expected value for the policy? Out comePayout xP(X=x) Death Disability Neither 1,000 500 0 1 / 1000 2 / 1000 997 / 1000

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Class Practice On a multiple-choice test, a student is given five possible answers for each question. The student receives 1 point for a correct answer and loses ¼ point for an incorrect answer. If the student has no idea of the correct answer for a particular question and merely guesses, what is the student’s expected gain or loss on the question? Suppose also that on one of the questions you can eliminate two of the five answers as being wrong. If you guess at one of the remaining three answers, what is your expected gain or loss on the question? (pg 5.1 - 5.5)

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