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Expected Value of a Discrete Random Variable Lecture 8B

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Expected Value of a Discrete Random Variable Lecture 8B

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    1. Expected Value of a Discrete Random Variable Lecture 8B William F. Hunt, Jr. Statistics 361 Section 5.4 (cont.)

    2. Center The mean of the probability distribution is the expected value of X, denoted E(X) E(X) is also denoted by the Greek letter µ (mu)

    3. k = the number of possible values (k=4) µ = x1·p(x1) + x2·p(x2) + x3·p(x3) + ... + xk·p(xk) Weighted mean Mean or Expected Value

    4. k = the number of outcomes (k=4) µ = x1·p(x1) + x2·p(x2) + x3·p(x3) + ... + xk·p(xk) Weighted mean Each outcome is weighted by its probability Mean or Expected Value

    5. Other Weighted Means Stock Market: The Dow Jones Industrial Average The “Dow” consists of 30 companies (the 30 companies in the “Dow” change periodically) To compute the Dow Jones Industrial Average, a weight proportional to the company’s “size” is assigned to each company’s stock price

    6. k = the number of outcomes (k=4) µ = x1·p(x1) + x2·p(x2) + x3·p(x3) + ... + xk·p(xk) EXAMPLE Mean

    7. k = the number of outcomes (k=4) µ = x1·p(x1) + x2·p(x2) + x3·p(x3) + ... + xk·p(xk) EXAMPLE µ = 10*.20 + 5*.40 + 1*.25 – 4*.15 = 3.65 ($ mil) Mean

    8. k = the number of outcomes (k=4) µ = x1·p(x1) + x2·p(x2) + x3·p(x3) + ... + xk·p(xk) EXAMPLE µ = 10·.20 + 5·.40 + 1·.25 - 4·.15 = 3.65 ($ mil) Mean

    9. Interpretation E(x) is not the value of the random variable x that you “expect” to observe if you perform the experiment once

    10. Interpretation E(x) is a “long run” average; if you perform the experiment many times and observe the random variable x each time, then the average x of these observed x-values will get closer to E(x) as you observe more and more values of the random variable x.

    11. Example: Green Mountain Lottery State of Vermont choose 3 digits from 0 through 9; repeats allowed win $500 x $0 $500 p(x) .999 .001 E(x)=$0(.999) + $500(.001) = $.50

    12. Example (cont.) E(x)=$.50 On average, each ticket wins $.50. Important for Vermont to know E(x) is not necessarily a possible value of the random variable (values of x are $0 and $500)

    13. Example p. 13 Suppose a fair coin is tossed 3 times and we let x=the number of heads. Find m=E(x). First we must find the probability distribution of x.

    14. Example (cont.) Possible values of x: 0, 1, 2, 3. p(1)? An outcome where x = 1: THT P(THT)? (½)(½)(½)=1/8 How many ways can we get 1 head in 3 tosses? 3C1=3

    15. Example (cont.)

    16. Example (cont.) So the probability distribution of x is: x 0 1 2 3 p(x) 1/8 3/8 3/8 1/8

    17. Example

    18. US Roulette Wheel Let x be your winnings on 1 play of a single number x -1 35 p(x) 37/38 1/38 E(x)= -1(37/38)+35(1/38)= -.05

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