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Unformatted text preview: M4056 Lecture Notes. Monday, September 27, 2010 General orientation. We have decided to model reality using a probability distribution from a parametrized family f X ( x | ), but we do not know the the value of that best describes reality. We wish to use the information in a random sample to select it. The likelihood function is defined as follows: L ( | vectorx ) := f vector X ( vectorx | ) . The likelihood function is the pdf (or pmf ) of the sample, viewed not a function of the sample value, but as a function of the parameter. Example 1. Suppose X is Bernoulli( p ), where p (the probability of success) is unknown. Consider the general sample vector X of n elements, and let vectorx be a specific observation. Then L ( p | vectorx ) = f vector X ( vectorx | p ) = p t (1 p ) n- t , where t := n i =1 x i . We see that L ( p ) = L ( p | vectorx ) is defined and non-negative on [0 , 1], with L (0) = 0 = L (1). We have dL dp = tp t- 1 (1 p ) n- t p t ( n t )(1 p ) n- t- 1 = p t- 1 (1 p ) n- t- 1 ( t pn ) , so L has a critical point at p = t/n , and this is a maximum. Note that integration by parts (...
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