Chapter 4

# Chapter 4 - IENG 213 Probability and Statistics for...

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Click to edit Master subtitle style IENG 213: Probability and Statistics for Engineers Instructor: Steven E. Guffey, PhD, CIH © 2002-2009 Chapter 4: Mathematical Expectation If you don’t know where you are going, you will probably end up somewhere else. Yogi Bera

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22 4.1 Mean of a Random Variable Definition 4.1: Let X be a random variable with probability distribution f(x). The mean or “expected value” of X is: = = x x f x X E ) ( ) ( μ dx x f x X E - = = ) ( ) ( if X is discrete if X is continuous
33 Example 4.1 f(0)= 1/35 f(2)= 18/35 f(1)= 12/35 f(3)= 4/35 = = x x f x X E ) ( ) ( μ

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44 Example 4.3 Let X be a random variable with density function: f(x) = 20,000 / x3, X>100 = 0, elsewhere x
55 Example 4.3 continued- get better example xxxx & = 200 hrs, from previous calculations Assign the random variable, g(X): g(X) = X2 , for x = -1, 0, 1, 2 x | 0 -1, 1 2

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66 Theorem 4.1 Let X be a random variable with a probability distribution f(x). The mean or expected value of the random variable g(X) is: [ ] = = x x g x f x g X g E ) ( ) ( ) ( ) ( μ [ ] dx x f x g X g E x g - = = ) ( ) ( ) ( ) ( if X is discrete if continuous
77 Example Car Sold Per Visit Fraction sold Income per sale luxury convertible 0.03 \$400 economy sedan 0.12 \$220 none 0.85 \$0 Given the sales data above for a car salesman, what is his expected income for each visit to the show room?

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88 Example The salesman’s manager gets a cut equal to 25% of the salesman after the first \$100 of each sale. What is her expected take per visit? Car Sold Per Visit Fraction sold Income per sale Mgr per sale luxury convertible 0.03 \$400 \$300/4 economy sedan 0.12 \$220 \$120/4 none 0.85 \$0 0 Manager’s cut = 0.25 x (Salesman-100)
99 Example The distribution below is for tips for one waiter throughout the evening. What is his expected earnings if he receives the following distribution of tips for his 12 customers? X \$4 \$5 \$6 \$7 \$8 \$9 f(x) 1/12 1/12 3/12 3/12 2/12 2/12 x f(x) 4/12 5/12 18/12 21/12 16/12 18/12 Total 12/12 82/12

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10 Example Another waiter earns g(x)=2x-1. How much will he earn over the evening if the distribution of x is as shown below? x 4 5 6 7 8 9 Total g(x) = 2x-1 7 9 11 13 15 17 72 f(x) 0.08 0.08 0.25 0.25 0.17 0.17 1.00 g(x) f(x) 0.56 0.75 2.75 3.25 2.50 2.83 12.7
1111 Joint Probability Functions and Expected Values ( 29 X E ( , ) x f x y dx dy ∞ ∞ -∞ -∞ = ∫ ∫ ( 29 E Y ( , ) y f x y dx dy ∞ ∞ -∞ -∞ = ∫ ∫

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12 Using Marginal Distribution to find E(X) Can use marginal distribution as shown below ( 29 X E ( , ) x f x y dx dy ∞ ∞ -∞ -∞ = ∫ ∫
13 HW: pp 113-115 Problems: No. 10, 12 plus all odd except 1, 3, 25, 27, 29, 31

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1414 Deviation from Mean Deviation = x- μ μ x Need more than the mean Deviation of an observation Difference between expected (i.e., average) and actual datum Deviation of an observation = x-&
1515 Deviation and Variance Deviation of an observation = x-& Variance = sum of squared deviations ( 29 dx x f x - - = ) ( 2 2 μ σ μ x Continuous ( 29 - = x x f X ) ( 2 2 Discrete 2 σ=

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1616 4.2 Expected value: the variance The variance of a random variable with a prob distribution f(x) and mean of & is: [ ] - = - = x x f X X E ) ( ) ( ) ( 2 2 2 μ σ if X is discrete ( 29 [ ] ( 29 dx x f x X E - - = - = ) ( 2 2 2 if continuous
1717 Example 4.8 Let the random variable X be the no. of cars used on any given workday. Show that the variance of B is greater than A if the prob. distribution is:

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Chapter 4 - IENG 213 Probability and Statistics for...

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