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Lecture 11-2007 - Lecture XI The Bernoulli distribution...

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Lecture XI
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The Bernoulli distribution characterizes the coin  toss.  Specifically, there are two events  X =0,1 with  X =1 occurring with probability  p .  The probability  distribution function  P [ X ] can be written as: Fall 2005 Lecture X 2 ( 29 1 [ ] 1 x x P X p p - = -
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Next, we need to develop the probability of  X + Y   where both  X  and  Y  are identically distributed.  If  the two events are independent, the probability  becomes: Fall 2005 Lecture X 3 [ ] [ ] [ ] ( 29 ( 29 ( 29 1 1 2 , 1 1 1 x y x y x y x y P X Y P X P Y p p p p p p - - - - + = = - - = -
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Now, this density function is only concerned with  three outcomes  Z = X + Y ={0,1,2}.  There is only one  way each for  Z =0 or  Z =2.  Specifically for  Z =0,  X =0  and  Y =0.  Similarly, for  Z =2,  X =1 and  Y =1.   However, for  Z =1 either  X =1 and  Y =0 or  X =0 or  Y =1.  Thus, we can derive: Fall 2005 Lecture X 4
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Fall 2005 Lecture X 5 ( 29 [ ] [ ] [ ] ( 29 ( 29 ( 29 [ ] ( 29 2 0 0 2 1 0 2 0 1 1 0 0 1 1 1 0 2 [ 0] 1 1 1, 0 0, 1 1 1 2 1 2 1 P Z p p P Z P X Y P X Y p p
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