1 / 11

Chapter 5 Discrete Probability Distributions

Chapter 5 Discrete Probability Distributions. 5.3 EXPECTATION 5.3.1 The Mean and Expectation (Expected Value) 5.3.2 Some Applications 5.4 VARIANCE AND STANDARD DEVIATION. 5.3 EXPECTATION. 5.3.1 The Mean and Expectation (Expected Value) Experimental approach

vidal
Download Presentation

Chapter 5 Discrete Probability Distributions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 5 Discrete Probability Distributions 5.3 EXPECTATION 5.3.1 The Mean and Expectation (Expected Value) 5.3.2 Some Applications 5.4 VARIANCE AND STANDARD DEVIATION

  2. 5.3 EXPECTATION • 5.3.1 The Mean and Expectation (Expected Value) • Experimental approach • Suppose we throw an unbiased die 120 times and record the results: • Then we can calculate the mean score obtained where = = = ________ (3 d.p.)

  3. Theoretical approach • The probability distribution for the random variable X where X is ‘the number on the die’ is as shown: • We can obtain a value for the ‘expected mean’ by multiplying each score by its corresponding probability and summing, so that Expected mean = =

  4. If we have a statistical experiment: • a practical approach results in a frequency distribution and a mean value, • a theoretical approach results in a probability distribution and an expected value. The expectation of X (or expected value), written E(X) is given by E(X) =

  5. Example 1 • random variable X has a probability function defined as shown. Find E(X).

  6. In general, if g(X) is any function of the discrete random variable X then In general, if g(X) is any function of the discrete random variable X then E[g(X)] =

  7. Example • In a game a turn consists of a tetrahedral die being thrown three times. The faces on the die are marked 1,2,3,4 and the number on which the die falls is noted. A man wins $ whenever x fours occur in a turn. Find his average win per turn.

  8. Example • The random variable X has probability function P(X = x) for x = 1,2,3. • Calculate (a) E(3), (b) E(X), (c) E(5X), (d) E(5X+3), (e) 5E(X) + 3, (f) E(X2), (g) E(4X2- 3), (h) 4E(X2 ) – 3. • Comment on your answers to parts (d) and (e) and parts (g) and (h).

  9. E(a X + b) = a E(X) + b, where a and b are any constants. E[f1(X)  f2(X)] = E[f1(X)]  E[f2(X)], where f1 and f2 are functions of X.

  10. 5.3.2 Some Applications

  11. 5.4 VARIANCE AND STANDARD DEVIATION The variance of X, written Var(X), is given by Var(X) = E(X - )2

More Related