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Chapter 4: Q.4.20 (page 174)
Show that if X is a geometric random variable with parameter p, then Hint: You will need to evaluate an expression of the form
to do so, write
and then interchange the sum and the integral.In the given information the answer is
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A man claims to have extrasensory perception. As a test, a fair coin is flipped times and the man is asked to predict the outcome in advance. He gets out of correct. What is the probability that he would have done at least this well if he did not have ESP?
In some military courts, judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability .whereas when the defendant is, in fact, innocent, this probability drops to .
(a) What is the probability that a guilty defendant is declared guilty when there are (i) , (ii) , and (iii) judges?
(b) Repeat part (a) for an innocent defendant.
(c) If the prosecuting attorney does not exercise the right to a peremptory challenge of a judge, and if the defense is limited to at most two such challenges, how many challenges should the defense attorney make if he or she is percent certain that the client is guilty?
If the distribution function of is given by
calculate the probability mass function of .
Three dice are rolled. By assuming that each of the possible outcomes is equally likely, find the probabilities attached to the possible values that X can take on, where X is the sum of the 3 dice.
Consider a random collection of individuals. In approximating the probability that no of these individuals share the same birthday, a better Poisson approximation than that obtained in the text (at least for values of between and ) is obtained by letting be the event that there are at least 3 birthdays on day
(a) Find .
(b) Give an approximation for the probability that noindividuals share the same birthday.
(c) Evaluate the preceding when (which can be shown to be the smallest value offor which the probability exceeds.).
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