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16 Mathematics of Managing Risks

16 Mathematics of Managing Risks. Weighted Average Expected Value. Weighted Average or Weighted Mean. The weighted average (or weighted mean ) of a set of N numbers each of which is assigned a weight where is:. Examples.

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16 Mathematics of Managing Risks

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  1. 16 Mathematics of Managing Risks • Weighted Average • Expected Value

  2. Weighted Average or Weighted Mean The weightedaverage (or weightedmean) of a set of N numbers each of which is assigned a weight where is:

  3. Examples If homework/quiz average is weighted 20%, 2 exams are weighted 25% each, and final exam is weighted 30% and a student makes homework/quiz average 87, exam scores of 80 and 92, and final exam score 85. Compute the weighted average.

  4. Examples The weightedaverage is

  5. Random Variable A random variable is a letter (X) that denotes a single numerical value which is observed when performing a random experiment.

  6. Examples of Random Variable • Toss a coin 3 times and countthe number of heads. Denote the total number of heads by the random variable X. • A basketball player shoots two consecutive free throws. Denote the total number of points scored by the random variable X.

  7. Probability Distribution A probability distribution for a random variable X gives the probability for any value of X. (Note: this is similar to a probability assignment for a sample space) Example: Toss a coin 3 times and countthe number of heads. Denote the total number of heads by the random variable X. What is the probability distribution for X?

  8. Probability Distribution

  9. Expected Value of a Random Variable The expected value (E) of a random variable X which has N possible outcomes each of which is assigned a probability where is:

  10. Expected Value of a Random Variable • The formula for the expected value is similar to a weighted average formula. • The expected value of a random variable X gives the approximate value of X that would result after repeating the random experiment many, many times.

  11. Example Toss a coin 3 times and countthe number of heads. Denote the total number of heads by the random variable X. What is the expected value of X? (Use the probability distribution in the previous example.)

  12. Example That is, we expect there will be 1.5 heads in three tosses (that is, we expect that 50% of the tosses would result in heads).

  13. Example page 621

  14. Example • X is a random variable that represents the net gain (or loss) of your bet. • Probability distribution of X is (assuming each guess equally likely):

  15. Example The negative indicates that if the random experiment were repeated many times, there would be a net loss of about $0.03 (house wins).

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