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Lecture (10)

Lecture (10). Mathematical Expectation. Mathematical Expectation. The expected value of a variable is the value of a descriptor when averaged over a large number theoretically infinite. Mathematical Expectation (cont.). Another way to compute the variance. 0 1 2. TT TH HT HH.

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Lecture (10)

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  1. Lecture (10) Mathematical Expectation

  2. Mathematical Expectation The expected value of a variable is the value of a descriptor when averaged over a large number theoretically infinite.

  3. Mathematical Expectation (cont.) Another way to compute the variance

  4. 0 1 2 TT TH HT HH Example 1 Sample Space Number of Heads

  5. Example 1 (cont.)

  6. Experiment: Toss Two Coins 1 Y T I L I 0 . 5 B A B O R P .25 0 1 2 3 N U M B E R O F H E A D S Example 1 (cont.)

  7. Example 1 (cont) • E.G. Toss 2 coins, count heads, compute expected value: • = 0  .25 + 1 .50 + 2  .25 = 1 E.G. Toss 2 coins, count heads, compute variance: variance = (0 - 1)2 (.25) + (1 - 1)2 (.50) + (2 - 1)2(.25) = .50

  8. Example 2

  9. Discrete Uniform Distribution Example • Find the mean of the number of spots that appear when a die is tossed. The probability distribution is given below.

  10. Discrete Uniform Distribution Example (cont.) That is, when a die is tossed many times, the theoretical mean will be 3.5.

  11. Binomial Distribution -Example • A coin is tossed four times. Find the mean, variance and standard deviation of the number of heads that will be obtained. • Solution:n = 4, p = 1/2 and q = 1/2. •  = np = (4)(1/2) = 2. • 2 = npq = (4)(1/2)(1/2) = 1. •  = = 1.

  12. Poisson Distribution

  13. Uniform Distribution Example

  14. Example • If the probability density function has the form • f(x) = ax for a random variable X between 0 and 2. • Find the value of a. • Find the median of X • Find P(1.0 < X < 2.0) Solution: (a) From the area under the whole density curve is 1, then we have

  15. Quiz

  16. Exponential Distribution

  17. Comparison of Parameters of Dist’n

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