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Unformatted text preview: Definition: Let X be a discrete random variable with probability distribution f(x). The mean , u, or expected value, E ( X ), of X is Definition: Let X be a continuous random variable with probability distribution f(x). The mean, or expected value , of X is 5/15/10 1 , 1 3 (ln3) ( ) 0, other values of x. x x f x ≤ ≤ = Verify that this is a valid pdf. 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.20.20.4 0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 f x ( 29 = 1 x ⋅ ln 3 ( 29 5/15/10 1 , 1 3 (ln3) ( ) 0, other values of x. x x f x ≤ ≤ = Find the mean, u , of this continuous pdf. 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.20.20.4 0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 f x ( 29 = 1 x ⋅ ln 3 ( 29 5/15/10 ( 29 3 1 1 ln 3 x dx x = ∫ 5/15/10 The distribution of the lifetime of a 9 volt alkaline battery (in 100’s of hours) is given by 2 14 4 ,0 3 27 ( ) 0, x x x l x otherwise  ≤ ≤ = Determine the expected lifetime for this type of battery. 5/15/10 5 (100) 166.6 hours. 3 B 5/15/10 Determining the expected value of functions of X . ( 29 1. ( ) ? Continuous case: ( ) ( ) ( ) = ( ) ( ) = ( ) ( ) ( ) 1 ( ) E a x b E ax b a x b f x dx a x f x dx b f x dx a x f x dx b f x dx a E x b a E x b ⋅ ± = ± = ⋅ ± = ⋅ ⋅ ± ⋅ ⋅ ± = ⋅ ± = ⋅ ± ∫ ∫ ∫ ∫ ∫ 5/15/10 Determining the expected value of functions of X . [ ] [ ] 1. ( ) ( ) 2. ( ) ( ) ( ), 3. ( ) ( ) ( ) , all x E ax b aE x b E g x g x f x discrete E g x g x f x dx continuous ∞∞ ± = ± = = ∑ ∫ 5/15/10 Determining the expected value of functions of X . ( 29 2 Suppose that 3 1 , 1 1 ( ) 4 0, other values of x x f x x  ≤ ≤ = Is this a valid density function? 5/15/10 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.10.21.210.80.60.40.2 0.2 0.4 0.6 0.8 1 1.2 ( 29 2 Suppose that 3 1 , 1 1 ( ) 4 0, other values of x x f x x  ≤ ≤ = 5/15/10 1.61....
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 Spring '10
 Wallace
 Probability, Variance, Probability theory, probability density function

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