l5 - Outline Discrete Distributions Oct 4, 2011 Discrete...

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Outline Discrete Distributions Oct 4, 2011 Discrete Distributions
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Outline Outline 1 Distributions pt 1 2 Distributions pt 2 Discrete Distributions
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Distributions pt 1 Distributions pt 2 Bernoulli Binomial Outline 1 Distributions pt 1 2 Distributions pt 2 Discrete Distributions
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Distributions pt 1 Distributions pt 2 Bernoulli Binomial Up to now, we’ve defined the pdf’s for our R.V.s by listing out all the possible values the R.V. could take and giving a probability for each outcome Rather cumbersome in real life What we’d love is if a small number of parameters in some function could define the probabilities for us E.g., Pr ( X = x | a , b ) where a and b are some fixed parameters We’ll spend the next few weeks talking about some common functions Discrete Distributions
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Distributions pt 1 Distributions pt 2 Bernoulli Binomial In many ways this is the simplest distribution. The Bernoulli distribution may be appropriate when there are only two possible outcomes (heads/tails; yes/no; dead/alive) and you only observe it once. Bernoulli pdf: X Bern ( p ) pr ( X = 0 ) = 1 - p pr ( X = 1 ) = p For the Bernoulli distribution the parameter is p is the probability of observing X = 1. The Bernoulli distribution serves as the basis of logistic regression Discrete Distributions
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Distributions pt 1 Distributions pt 2 Bernoulli Binomial Bernoulli PDF and CDF Discrete Distributions
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Distributions pt 1 Distributions pt 2 Bernoulli Binomial E [ X ] , V [ X ] E [ X ] = X x xpr ( x ) = 1 × p + 0 × ( 1 - p ) = p V [ X ] = X x ( x - p ) 2 pr ( x ) = ( 0 - p ) 2 ( 1 - p ) + ( 1 - p ) 2 p = p 2 ( 1 - p ) + p ( 1 - p ) 2 = p ( 1 - p ) Discrete Distributions
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Distributions pt 1 Distributions pt 2 Bernoulli Binomial Suppose we want to model some disease using the Bernoulli distribution D i Bernoulli ( p ) , where D i is disease status for individual i . What are we assuming about the disease? Outcomes are independent in the sense that the i th person’s disease status doesn’t effect the j th person’s disease status Risk of disease ( p ) is homogenous Discrete Distributions
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Distributions pt 1 Distributions pt 2 Bernoulli Binomial Rather than a single trial, we might have n trials: x 1 ,..., x n . Each R.V. x i follows a Bernoulli dist’n. We might be interested in the R.V. Y = i x i . Binomial Distribution Pr ( Y = y ) = ± n y ² p y ( 1 - p ) n - y Discrete Distributions
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Distributions pt 2 Bernoulli Binomial Its easy to justify the binomial dist’n intuitively. Say n = 2 so we have the sample space S = { x 1 = 1 , x 2 = 1 } , { x 1 = 1 , x 2 = 0 } , { x 1 = 0 , x 2 = 1 } , { x 1 = 0 , x 2 = 0 } . Further, let x i Bern ( p ) . x 1 x 2 y Pr ( X 1 = x 1 ) Pr ( X 2 = x 2 ) Pr ( x 1 , x 2 ) 1 1 2 p p p 2 1 0 1 p 1 - p p ( 1 - p ) 0 1 1 1 - p p p ( 1 - p ) 0 0 0 1 - p 1 - p ( 1 - p ) 2 If we only care about Y = x i , then { x 1 = 1 , x 2 = 0 } and { x 1 = 0 , x 2 = 1 } both yield y = 1 and we don’t need to distinguish between them. We count the number of times that Y = y out of n
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l5 - Outline Discrete Distributions Oct 4, 2011 Discrete...

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