Example_class_6_handout_solution

Example_class_6_handout_solution - THE UNIVERSITY OF HONG...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 6 Review Some Common Continuous Distributions Gamma Function and Beta Function Γ( α ) = ´ 0 e - x x α - 1 dx, α > 0 Beta ( α,β ) = ´ 1 0 x α - 1 (1 - x ) β - 1 dx = Γ( α + β ) Γ( α )Γ( β ) 0.1 Uniform Distribution Let X U ( a,b ) , where b > a. The probability density function(pdf) of X is f ( x ) = ( 1 b - a if x [ a,b ] , 0 otherwise μ = a + b 2 σ 2 = ( b - a ) 2 12 0.2 Exponential Distribution Let X Exp ( λ ) , where λ > 0 . The PDF of X is f ( x ) = ( λe - λx if x = 0 0 otherwise μ = 1 λ σ 2 = 1 λ 2 A exponential random variable can be used to discribe random time elapsing between unpredictable events. 0.3 Gamma Distribution Let X Γ( α,λ ) , where α,λ > 0 . The pdf of X is f ( x ) = ( λ α x α - 1 e - λx / Γ( α ) if x > 0 0 otherwise μ = α λ σ 2 = α λ 2 Exponential distribution is a special case of Gamma distribution with α = 1 . Chi squared distribution is a special case of Gamma distribution of Γ(0 . 5 , 0 . 5) . 0.4 Chi-squared Distribution Let X χ 2 r , where r ( a positive integer) is the degree of freedom. The pdf of X is f ( x ) = ( x r/ 2 - 1 e - x/ 2 / (2 r/ 2 Γ( r/ 2)) if x > 0 0 otherwise μ = r σ 2 = 2 r 1
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0.5 Normal Distribution Let X N ( μ,σ 2 ) , where μ ( -∞ , ) is the mean and σ 2 > 0 is the variance. The Pdf of X is: f ( x ) = 1 p (2 πσ 2 e - ( x - μ ) 2 2 σ 2 , and -∞ < x < Some important points: (a) A normal distribution is symmetric about its mean. That is, if X N ( μ,σ 2 ) , then P ( X 6 μ - x ) = P ( X > μ + x ) In particular, Φ( x ) = Φ(1 - x ) , where Φ( x ) is the cumulative distribution function. (b)Transformation into standard normal Z (0 , 1) Z - score = x - μ σ (c)Relationship between normal and chi-squared distribution If X N (0 , 1) , then X 2 χ 2 1 If Z 1 ,...,Z k are independent random variables, each with a standard normal distribution, then by definition Z 2 1 + · · · + Z 2 k has a χ 2 k distribution. 0.6 Beta Distribution
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Example_class_6_handout_solution - THE UNIVERSITY OF HONG...

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