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Unformatted text preview: Lecture Oct 6 CORRECTED!! Last time looked at uniform distribution and exponential distribution Today get after gamma distribution Background—the gamma function. This is a way to extend n! to all positive numbers. Probably new to you—not covered in calculus For α > 0, define Γ (α ) = xα −1 exp ( − x ) dx Who would come up with this? Comes up in 19th century ∞ ∫
0 applied math (precomputer applied math) a lot. Learn about it. Properties: Γ (1) = 1. (easy verification) Γ (α + 1) = αΓ (α ) (integrate by parts—we do this at board) Corollary: for any positive integer n, Γ ( n ) = ( n − 1) ! Another fact: Γ (1 2 ) = π not that hard to see‐but we want to do this later in connection with other stuff. Example: Find Γ ( 7 2 ) ANS: ( 5 2 ) ( 3 2 ) (1 2 ) π Formula: xα −1 exp ( −λ x ) dx =
∞ ∫
0 Γ (α ) λα ∞ ∞ Proof: Let y = λ x. then dx = dy λ and so xα −1 exp ( −λ x ) dx =λ −α yα −1 exp ( − y ) dy = ∫
0 ∫
0 Γ (α ) λα THEREFORE for fixed α > 0, λ > 0 f ( x ) = DENSITY with parameters α , λ λ α xα −1 exp ( −λ x ) , x > 0, defines a density—the GAMMA Γ (α ) IMPORTANT SPECIAL CASES: 1. α = 1 THEN WE GET THE EXPONENTIAL DENSITY 2. α = n 2, λ = 1 2. THEN (WE WILL SEE) WE GET THE “CHI‐SQUARED DENSITY WITH N DEGREES OF 2 2 FREEDOM” IE THE DENSITY OF Z12 + Z 2 + … + Z n where the Z i ' s are independent standard normal random variables –a very important distribution for statistics. i.e. the gamma family of densities is a flexible family of densities on the positive real numbers which include the exponential density and the chi‐square density. A way to study all these densities at one time. Later in Chapter 2 we will see that if Z is standard normal, then Z 2 has the chi squared distribution with 1 degree of freedom, i.e. α = 1 2 and λ = 1 2. Still later in the course we will see that if Z1 , Z 2 ,… , Z n are independent standard normal, then the random variable Z12 + Z 22 + … + Z n2 has the chi squared distribution with n/2 degrees of freedom ie α = n 2 and λ = 1 2. Now onto the normal distribution: Background: Cute calculation from second year calculus (due to famous mathematician Liouville) ∞ −∞ ∫ exp ( − z 2 ) dz =
2 2π . Proof: Square the integral and hope we get 2π . In fact 2 ⎛∞ ⎞ ⎛∞ ⎞ ⎛∞ ⎞ 2 2 2 ⎜ ∫ exp ( − z 2 ) dz ⎟ = ⎜ ∫ exp ( − z 2 ) dz ⎟ ⋅ ⎜ ∫ exp ( − z 2 ) dz ⎟ ⎝ −∞ ⎠ ⎝ −∞ ⎠ ⎝ −∞ ⎠
∞ = −∞ 2 2 ∫ exp ( − x 2 ) dx ⋅ ∫ exp ( − y 2 ) dy since we can use any variable we want as a variable of −∞ ∞ integration. In the last product, move the first integral into the second (we can move constants inside ⎛∞ ⎞ 2 2 integrals) to get ∫ ⎜ ∫ exp ( − x 2 ) dx ⎟ exp ( − y 2 ) dy, and then move the exp ( − y 2 2 ) term inside −∞ ⎝ −∞ ⎠
∞
∞∞ the inner integral and combine exponentials to get −∞ −∞ ∫ ∫ exp ( − ( x 2 + y 2 ) 2 dxdy, Bottom line so ) far—the square of the integral can be written as a double integral. Now from doing double integral problems in second year calculus we think, since the integrand depends only on x 2 + y 2 , that maybe we should switch to polar coordinates. Let’s try that. We replace x 2 + y 2 by r 2 and replace dxdy by rdrdθ and fix the limits of integration to get that the double integral can 2π ∞ be written 2 ∫ ∫ exp ( − r 2 ) rdrdθ . The magic has occurred—while exp ( − z 2 ) cannot be integrated, 2 2 00 r exp ( − r 2 ) can be integrated. In fact the inside integral is easily seen to be 1, so that the double integral is 2π . This does the calculation. ⎛ ⎛ x − μ ⎞2 ⎞ Corollary: For any real μ and any σ > 0, ∫ exp ⎜ − ⎜ 2 ⎟ dx = 2π σ . ⎜⎝σ⎟ ⎟ ⎠ −∞ ⎝ ⎠
∞ Pf: Make the change of variable z = x−μ σ . ⎛ ⎛ x − μ ⎞2 ⎞ exp ⎜ − ⎜ 2⎟ ⎜⎝σ⎟ ⎟ ⎠ ⎝ ⎠ . Definition: The normal density with parameters μ , σ is f ( x ) = 2πσ
Notes: 1. We checked above this is a density (integrates to 1). 2. Down the road, after we define mean and standard deviation, we will check that the mean and standard deviation of this density are μ , σ respectively. 3. Since this density is symmetric about μ it is obvious that the median of the density is μ . ...
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This note was uploaded on 06/12/2010 for the course MATH 307 taught by Professor Luikonnen during the Fall '08 term at Tulane.
 Fall '08
 Luikonnen
 Calculus, Probability

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