Chapter 5 Discrete Probability Distributions

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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

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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

Scroe, x 1 2 3 4 5 6
Frequency, f 15 22 23 19 23 18



________ (3 d.p.)
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• 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

Score, x 1 2 3 4 5 6
P(X x) 1/6
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• 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)
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• Example 1
• random variable X has a probability function
  defined as shown. Find E(X).

-2 -1 0 1 2
P(X x) 0.3 0.1 0.15 0.4 0.05
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• 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
Eg(X)
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• 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.

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• 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(5X3),
• (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).

x 1 2 3
P(X x) 0.1 0.6 0.3
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E(a X b) a E(X) b, where a and b are any
constants.
Ef1(X) ? f2(X) Ef1(X) ? Ef2(X), where f1
and f2 are functions of X.
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• 5.3.2 Some Applications

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