TA4 - ECO2121: Methods of Economic Statistics TA4 – 11...

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Unformatted text preview: ECO2121: Methods of Economic Statistics TA4 – 11 February 2009 1. What probability distributions/random variables model? Discrete: Hypergeometric, Binomial, Poisson, Bernoulli, Geometric Continuous: Uniform, Exponential, Normal 2. Some important results (a) (revisit) If X has the mean μ and the variance σ2, X1,…,Xn are an i.i.d. sample of X, and denote as the sample mean, then has the mean μ and the variance 2 σ /n. (The interpretation is VERY important.) (b) If two events A and B are independent, show that their complements Ac and Bc are independent. How about A and Bc, as well as Ac and B? (How to interpret?) 3. Uniform Distribution X ~ Uniform(a, b) pdf f(x) = 1/(b – a), for a<x<b = () = − = 1 − = ( − )( + ) 1 + × = − 2 2 [ ]= − ( ) , so we first calculate EX2. 1 − = × () − 3 = = 1 − − × ( − )( + 2 = 1 − + 3 + = 1 − 1 − 2 = 1 − × − 2 = ∴ = = + ) √12 You are asked to derive the mean and variance of a random variable with exponential distribution with the parameter θ in Problem Set 3. Hence, 4. Exponential Distribution X ~ Exp(θ) ( )= 1 , >0 + [ ]= −2 12 + [ ]= = 3 (−) 12 − + − = 4( + = ) − 3( 12 + 3 3 + + +2 ) / / / =− −1 =1− , >0 We call this the distribution function of X, denoted as FX(x). Hence, Pr{X>a} = 1 – F(a) = 1 – (1 – e-a/θ) = e-a/θ ( )= { ≤ }= () = 1 / =− / = −[ / By this, we find an important property of the Exponential distribution, that is, for any b>a, Pr{X>b|X>a} = Pr{X>b∩X>a} / Pr{X>a} = Pr{X>b} / Pr{X>a} = e-b/θ / e-a/θ = e-(b–a)/θ, i.e. the same as the probability as if X is between the interval (a, b), it is usually called the “memory-less” property of the Exponential distribution. Taylor 10 February 2009 ...
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This note was uploaded on 11/16/2009 for the course ECO 2121 taught by Professor Professorwen during the Fall '08 term at Al Ahliyya Amman University.

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