HW32Answers(11.4)

HW32Answers(11.4) - j th average with j { 1 , 2 , . . . , m...

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Section 11.4 HW Assignment Math331, Spring 2008 Instructor: David Anderson Hw: Consider a biased coin (say prob. of heads is p > 1 / 2) being Fipped many times. Let X i be one if a head appears on i th Fip. Give a short write-up (roughly two paragraphs) interpreting both the weak and strong laws of large numbers in this case. Interpretation: Let X i be one if a head appears on i th Fip. Let ¯ X n = 1 n n s i =1 X i . Thus, ¯ X n represents the average of n Fips. Let ¯ X n ( j ) be the j th average of n Fips. That is, we perform the experiment of Fipping n coins many times, say m , and take ¯ X n ( j ) to be the
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Unformatted text preview: j th average with j { 1 , 2 , . . . , m } . We should think of both n and m as large. Then the weak law of large numbers says that for any > 0 ( is small), the probability that very many of the X n ( j ) are more than away from p , will be very small. urther, for xed , as n , the proportion that are more than away from p will go to zero. The Strong law of large numbers says that with a probability of one, X n ( j ) will converge to p as n . 1...
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