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Lecture12_print - MA/STAT416 Probability Lecture Notes...

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MA/STAT416: Probability Lecture Notes Spring 2011 1 Chapter 8: Special Continuous Distributions April 11, 2011 8 Special Densities 8.1 The Uniform Distribution Definition. Let X be a continuous random variable, the Uniform distribution with parameters a, b ; X U [ a, b ], is given by the following pdf f ( x ) = 1 b - a , a x b. Theorem. Suppose X U [ a, b ]. Then, μ X = E ( X ) = a + b 2 , and σ 2 X = V ar ( X ) = ( b - a ) 2 12 . Example 1 . Let X U [ a, b ]. If its mean is 2 and variance is 3, what are the values of a and b ? 8.2 The Exponential Distribution Definition. Let X be a continuous random variable, the Exponential distribution with parameter λ > 0; X Exp ( λ ), is given by the following pdf f ( x ) = 1 λ e - x λ , x > 0 . Definition. The Gamma function is defined by Γ( α ) = Z 0 x α - 1 e - x dx, α > 0 . Remark. In particular, Γ( n ) = ( n - 1)! , n = 1 , 2 , 3 , · · · . Γ( α + 1) = α · Γ( α ) , α > 0 . Γ 1 2 = π. Theorem. Suppose X Exp ( λ ). Then, μ X = E ( X ) = λ, and σ 2 X = V ar ( X ) = λ 2 . Example 2 . Suppose X Exp ( λ ). Answer the following: 1. Find the CDF of X ; i.e. F ( x ). 2. Evaluate the k th moment of X for a general k . 3. Evaluate P ( X > x + y | X > x ). This shows that the Exponential distribution also has the Lack of Memory property . Example 3 . A Two Layered Problem. The time that you have to wait to speak to a customer service representative when you call a bank is Exponentially distributed with mean 1.5 minutes. If you make 10 calls in one month (and never hang up), what is the
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