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Understanding Expected Value of a Random Variable in Probability Theory, Summaries of Statistics

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

  • How is expected value calculated?
  • What is the definition of expected value of a random variable?
  • What is the relationship between expected value and fair games?

Typology: Summaries

2021/2022

Uploaded on 09/12/2022

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7
Expected Value, E(X), of a Random Variable X
Start with an Experiment.
List the Outcomes:
O1, O2, … 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.
8
Calculating Expected Value
()
-
73 15
111 2
E = + + =
424 4
  
  
  
Make a table like this one
Outcome Probability Value of X
HH 1/4 7
HT or TH 1/2 3
TT 1/4 -15
O
1
O
2
O
3
X
1
X
2
X
3
pf3
pf4
pf5

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7

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.

8

Calculating Expected Value

E = + + =

Make a table like this one

Outcome Probability Value of X HH 1/4 7 HT or TH 1/2 3

TT 1/4 -

O 1

O 2

O 3

X 1

X 2

X 3

9

Fair Games; Expected Value is 0

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?

E D

Set 0

E

D

10

Expected Value - Example

  • The game costs $2 to play. You are dealt a poker hand. If it contains an Ace you get your $2 back, plus another $1. What is the (expected) value of the game to you?

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

13

Insurance Example

  • An insurance company charges $150 for a policy that will pay for at most one accident. For a major accident, the policy pays $5000; for a minor accident, the policy pays $1000. The $150 premium is not returned.
  • The company estimates that the probability of a major accident is 0.005, and the probability of a minor one is 0.08.
  • What is the expected value of the policy to the insurance company?

14

Insurance Example - 2

Outcomes Probability Cost to Premium company major accident

minor accident

no accident

E = 0.005(-$5000) + 0.08(-$1000) +0.915($0)+ $150 = $

15

Multiple Choice Tests

(a) (b) (c) (d) (e)

“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.

16

100 Questions Multiple Choice Test–

5 foils, different scoring

“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