STAT 333 probdists-midterm1

STAT 333 probdists-midterm1 - 1 12 ( b-a ) 2 m.g.f. M X ( t...

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1. Binomial R.V.: X BIN ( n,p ) range { 0 , 1 ,...,n } p.m.f f ( k ) = P ( X = k ) = ( n k ) p k (1 - p ) n - k mean E ( X ) = np variance V ar ( X ) = np (1 - p ) p.g.f. G X ( s ) = (1 - p + ps ) n for s R 2. Negative Binomial R.V.: Y NB ( r,p ) range { r,r + 1 ,r + 2 ,... } p.m.f f ( k ) = P ( Y = k ) = ( k - 1 r - 1 ) (1 - p ) k - r p r mean E ( Y ) = r p variance V ar ( Y ) = r (1 - p ) p 2 p.g.f. G Y ( s ) = ± ps 1 - s (1 - p ) ² r for | s | < 1 1 - p 3. Poisson R.V.: X POI ( λt ) range { 0 , 1 , 2 , 3 ,... } p.m.f f ( k ) = P ( X = k ) = e - λt ( λt ) k k ! mean E ( X ) = λt variance V ar ( X ) = λt p.g.f. G X ( s ) = e λt ( s - 1) for s R 4. Uniform R.V.: X UNIF ( a,b ) range ( a,b ) R p.f f ( x ) = 1 b - a c.d.f F ( x ) = x - a b - a mean E ( X ) = b + a 2 variance V ar ( X ) =
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Unformatted text preview: 1 12 ( b-a ) 2 m.g.f. M X ( t ) = e tb-e ta t ( b-a ) for t 6 = 0 , and M X ( t ) = 1 for t = 0 . 5. Gamma R.V.: X GAMMA ( , ) range [0 , ) p.f f ( x ) = x -1 e-x ( ) mean E ( X ) = variance V ar ( X ) = 2 m.g.f. M X ( t ) = -t for t &lt; 1...
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