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The concept of expected value of a random variable through an experiment with outcomes, probabilities, and values. It covers calculating expected value, fair games, and the relationship between expected value and winnings or losses. Examples include poker hands and insurance policies.
What you will learn
Typology: Summaries
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Expected Value, E(X), of a Random Variable X
Start with an Experiment. List the Outcomes : O 1 , O 2 , … On With each outcome is associated a probability: p 1 , p 2 , …, p (^) n and a value of the random variable, X, : X 1 , X 2 , …, X (^) n
The expected value of the random variable X is, by definition: E(X) = p(O 1 )X 1 + p(O 2 )X 2 +p(O 3 )X 3 + …. +p(O (^) n )X (^) n
The expected value is often denoted just by E.
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Make a table like this one
Outcome Probability Value of X HH 1/4 7 HT or TH 1/2 3
TT 1/4 -
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In a fair game, E = p(win)winnings +p(lose)loss = 0
Example: Suppose for some game, p(win) = 2/6; p(lose) = 4/ If you lose, you pay $1; if you win other player pays you $D What should D be if the game is to be fair?
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p(no Ace) = 0.659; p(at least one A) = 0.
Outcomes Probability Payoff to you
At least one ace
No aces 0.659 -$
E = 0.341*($1) + 0.659(-$2) = -$0.978 (^) value of game to player
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Outcomes Probability Cost to Premium company major accident
minor accident
no accident
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“Your grade = # of correct answers - (1/4)(# of incorrect answers)”
Suppose you guess at the answer to a question. What is the expected number of points you’ll get for that question?
Outcome Probability Value
guess right guess wrong
1 point
-(1/4) point
E = (1/5)1 + (4/5)(-1/4) = 0
Or, you get 1 point for each correct answer, and –(1/4)pt for each incorrect answer.
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“Your grade = # of correct answers - (1/5)(# of incorrect answers)” Suppose you guess at the answer to all 100 questions. What is the expected grade for the test? Per question: Outcome Probability Value
guess right
guess wrong
1 point
-(1/5) point
E = (1/5)1 + (4/5)(-1/5) = 0.
For the test: 100*0.04 = 4