STAT312 Lecture Notes - Exponential Distribution, Independent And Identically Distributed Random Variables, Electronic Component

• School: University of Alberta
• Department: Statistics
• Course: STAT312
• Professor: Douglas Wiens

Document Summary

P( ) (poisson, mean (for this reason the. Poisson is sometimes known as the distribution of the number of rare events . ) Proof: first calculate ( ) = x . We aim to show (why?) that this as. Take logs: log 1 log 1 + log 1 + . Example 2: suppose exponentially distributed with density. 1 are i. i. d. , each: the m. g. f. of each is ( ) =z0. We will see later how to obtain the density of (by induction for instance); it will turn out to be the erlang den- sity. Now we have an easy proof of this: h i = 1. We know that if we can ex- pand the m. g. f. as ( ) = x=0: using (24. 1) we get, for | | = h ( ) = x=0 + then whence. Example 4: if ( ) = h stants, then =p =1 i = = h. 2 2 as in example 2, by the clt.