Example Class 5 Solution - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 5 Review 1. Expectation Assume that E ( X ) and E ( g ( X )) exist in the following, then E ( X ) E ( g ( X )) X is a discrete r.v. ∑ Ω ∈ ) ( ) ( X x x xp ∑ Ω ∈ ) ( ) ( ) ( X x x p x g X is a continuous r.v. ∫ Ω ) ( ) ( X dx x xf ∫ Ω ) ( ) ( ) ( X dx x f x g Mean of X = E ( X ) Mean and variance Variance of X = Var ( X ) = ( ) 2 )) ( ( X E X E − = E ( X 2 ) – E ( X ) 2 The mean is usually denoted by µ and the variance is usually denoted by σ 2 . 2. Moments and moment generating function Let r be a positive integer. The r th moment of X is E ( X r ). The r th moment of X about b is Moment ( ) r b X E ) ( − . Let X be a random variable. The moment generating function of X is defined as Moment generating function M X ( t ) = E ( e tX ) if it exists. The domain of ( ) are all real number t such that the expectation above is finite. And we have ( ) (0) = ¡ ¢ £ ( ¤ ) ¤ ¡ ¤=0 = E ( X r ) if M X ( t ) is differentiable at t = 0. Remark: moment generating function uniquely characterizes the distribution. 3. Some common continuous distributions Uniform distribution. Let X ~ U ( a , b ), where b > a . The probability density function (pdf) of X is f ( x ) = 1 − if ∈ [ ¡ , ¢ ], otherwise. Exponential distribution. Let X ~ Exp ( λ ), where λ > 0. The pdf of X is f ( x ) = otherwise. , if > − x e x λ λ Gamma distribution. Let X ~ Gamma( α , λ ), where α , λ > 0. The pdf of X is f ( x ) = otherwise. , if ) ( / 1 > Γ − − x e x x α λ λ α α Chi-square distribution. Let X ~ 2 r χ , where r (a positive integer) is the degrees of freedom. The pdf of X is f ( x ) = otherwise. , if )) 2 / ( 2 /( 2 / 2 / 1 2 / > Γ − − x r e x r x r 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 ) = ) 2 /( ) 2 /( ) ( 2 2 π σ σ µ − − x e , – ∞ < x < ∞ . Beta distribution. Let X ~ Beta ( α , β ), where α , β > 0. The pdf of X is f ( x ) = otherwise. , 1 if ) 1 ( ) ( ) ( ) ( 1 1 < < − Γ Γ + Γ − − x x x β α β α β α 4. Markov inequality and Chebyshev’s inequality Markov’s inequality. If X is a non-negative random variable with finite mean E ( X ), then for any constant c > 0, ( ≥ ) ≤ ( £ ) ¤ . Chebyshev’s inequality. If the random variable X has finite mean µ and finite variance 2 , then for any real number k > 0, (| ¥ | ≥ ) ≤ 1 ¦ § . Problems Problem 1. Let X follow a discrete uniform distribution on [ a , b ], where a and b are integers with a ≤ b . The pmf of X is P ( X = x ) = p ( x ) = 1/( b – a + 1) if a ≤ x ≤ b and p ( x ) = 0 otherwise....
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This note was uploaded on 11/17/2011 for the course MUSIC 1001 taught by Professor Dr.yun during the Spring '11 term at Princess Sumaya University for Technology.

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Example Class 5 Solution - THE UNIVERSITY OF HONG KONG...

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