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Probability : Probability Generating Function and Beta and Gamma Function

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3.2 A majority verdict of 10-2 or better may be permitted in a jury trial. Assuming each juror has probability 0.9 of reaching a Guilty verdict, and decides independently, what is the probability the jury decides to convict?
(Partial solution provided in attachment)

3.14 For both the Gamma and Beta distribution, find the value of x that maximizes the density function. When is there just one local maximum?
(Partial solution provided in attachment)

3.8 The Logarithmic distribution has been used in ecology to model how often any particular species appears in a habitat. Its pgf is of the form cln(1 - qz), where c is a constant and 0 < q < 1. Find c in terms of the parameter q, and write down the expression fo pk, the chance a species is located k times.
(Partial solution provided in attachment)

3.12 Eleven horses enter a race. Mr C takes exactly 60 second to complete the course. Each of the other ten horses, independently, takes more than 60 seconds with probability p, or less than 60 seconds with probability 1 - p. Give a model for Mr C's position in the race. Find the chances that he wins, and he is placed (i.e. in the first three). Find the value of p that makes all eleven horses equally likely to win. For this value of p, find the chance Mr C is placed. Is it a surprise that your last answer is not 3/11?
(Partial solution provided in attachment)

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A Probability Generating Function and Beta and Gamma Function are investigated. General Probability problems are solved. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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3.2 A majority verdict of 10-2 or better may be permitted in a jury trial. Assuming each juror has probability 0.9 of reaching a Guilty verdict, and decides independently, what is the probability the jury decides to convict?

The probability of n jurors voting guilty is because n will vote guilty at 0.9 probability, 12 - n will vote not guilty and the n jurors can be chosen in 12Cn ways. Since we need at least 10 jurors to vote ...

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