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GE 331-Lecture 23

# GE 331-Lecture 23 - Final Time Monday 1:30PM to 4:30PM...

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Final Time: 5/11/2009 Monday 1:30PM to 4:30PM Location: 1DCL-1310 1DCL-1320

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Quiz 7 A biased coin is tossed 80 times. Let p denote the probability of observing a Head. Suppose that the toss resulted in 49 HEADS and 31 TAILS. Find the maximum likelihood (ML) estimator of p. (Make sure you explain how you reached your decision to receive full credit)

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The joint probability density function , , X Y f u v for the continuous random variables X and Y has constant value on the shaded region : {(u, v) : 0 < u < 2, 0 < v < 2, 1 < u + v < 2} (a) Find   X f u , the marginal probability density function for X . (b) Find | E Y X x , the conditional mean estimate of Y given X has been observed. Problem 1
u v u+v=2 u+v=1 1 1 2 2 0

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u v u+v=2 u+v=1 dvdu v from 1-u to 2-u, u from 0 to 1 dvdu v from 0 to 2-u, u from 1 to 2 1 1 2 2 0 Region I Region II
find the joint pdf, we know it is a constant, then assume , , X Y f u v c , then we have , I and II I and II 1 2 2 2 0 1 1 0 , 3 3 1 2 1 2 2 2 3 X Y region region u v u u v u u v u u v f u v dvdu cdvdu cdvdu cdvdu c c c         Thus we have , 2 ,0 2,0 2,1 2 , 3 0, X Y u v u v f u v otherwise

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u v u+v=2 u+v=1 dudv u from 0 to 2-v, u from 1 to 2 dudv u from 1-v to 2-v, v from 0 to 1 1 1 2 2 0 Region I Region II
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• Spring '09
• NegarKayavash
• Probability theory, Exponential distribution, probability density function, continuous random variable, dvdu

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GE 331-Lecture 23 - Final Time Monday 1:30PM to 4:30PM...

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