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Lecture11-2010 - Binomial and Normal Random Variables...

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Binomial and Normal Random Variables Charles B. Moss July 14, 2010 I. Bernoulli Random Variables A. The Bernoulli distribution characterizes the coin toss. Specifically, there are two events X = 0 , 1 with X = 1 occurring with probabil- ity p . The probability distribution function P [ X ] can be written as P [ X ] = p x (1 p ) x (1) B. Next, we need to develop the probability of where both and are identically distributed. If the two events are independent, the probability becomes P [ X, Y ] = P [ X ] P [ Y ] = p x p y (1 p ) 1 - x (1 p ) 1 - y = p x + y (1 p ) 2 - x - y (2) C. Now, this density function is only concerned with three outcomes Z = X + Y = { 0 , 1 , 2 } . 1. There is only one way each for Z = 0 or Z = 2. a) Specifically for Z = 0 , X = 0 and Y = 0. b) Similarly, for Z = 2 , X = 1 and Y = 1. 2. However, for Z = 1 either X = 1 and Y = 0 or X = 0 and Y = 1 . Thus, we can derive 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XI Fall 2010 P [ Z = 0] = p 0 (1 p ) 2 - 0 P [ Z = 1] = P [ X = 1 , Y = 0] + P [ X = 0 , Y = 1] = p 1+0 (1 p ) 2 - 1 - 0 + p 0+1 (1 p ) 2 - 0 - 1 = 2 p 1 (1 p ) 1 P [ Z = 2] = p 2 (1 p ) 0 (3) D. Next we expand the distribution to three independent Bernoulli
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  • Spring '10
  • Staff
  • Probability distribution, Probability theory, Cumulative distribution function, econometric methods, Lecture XI, Professor Charles B. Moss

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