{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Example_class_6_handout_solution

# Example_class_6_handout_solution - THE UNIVERSITY OF HONG...

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 9

Example_class_6_handout_solution - THE UNIVERSITY OF HONG...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online