notes3 - MTH4106 Introduction to Statistics Notes 3 Spring...

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Unformatted text preview: MTH4106 Introduction to Statistics Notes 3 Spring 2012 Discrete random variables Some revision If X is a discrete random variable then X may take finitely many values x 1 < x 2 < < x n or infinitely many values { x i : i Z } so long as no two are too close together . Write p i = P ( X = x i ) = probability that X = x i . The list of the p i is called the probability mass function , and i p i = 1 . The expectation of X is E ( X ) = i p i x i . If g is any real function, then g ( X ) is a random variable and E ( g ( X )) = i p i g ( x i ) . In particular, E ( X 2 ) = i p i x 2 i and the variance of X is Var ( X ) = E ( X 2 )- [ E ( X )] 2 . 1 Bernoulli random variable Given p in ( , 1 ) , we say that X Bernoulli ( p ) if P ( X = ) = q and P ( X = 1 ) = p , where q = 1- p . Binomial random variable Given p in ( , 1 ) and a positive integer n , we say that X Bin ( n , p ) if P ( X = i ) = n i q n- i p i for i Z with 0 6 i 6 n , where q = 1- p . Geometric random variable Given p in ( , 1 ) , we say that X Geom ( p ) if P ( X = i ) = q i- 1 p for all positive integers i , where q = 1- p . Hypergeometric random variable Suppose that we have N sheep in a field, of which M are black and the rest white. We sample n sheep from the field without replacement . Let the random variable X be the number of black sheep in the sample. Such an X is called a hypergeometric random variable Hg ( n , M , N ) . Here n , M and N are positive integers with n 6 N and M 6 N . P ( X = i ) = ( M i )( N- M n- i ) ( N n ) for 0 6 i 6 n ....
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notes3 - MTH4106 Introduction to Statistics Notes 3 Spring...

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