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Chapter 4: Q.4.18 (page 171)
Let be a Poisson random variable with parameter . What value of maximizes
The solution is.
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In the game of Two-Finger Morra, players show or fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither guesses correctly, then no money is exchanged. Consider a specified player, and denote by X the amount of money he wins in a single game of Two-Finger Morra.
(a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the possibilities is equally likely, what are the possible values of and what are their associated probabilities?
(b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up or fingers, what are the possible values of and their associated probabilities?
Consider n independent trials, each of which results in one of the outcomes with respective probabilities Show that if all the are small, then the probability that no trial outcome occurs more than once is approximately equal to.
When coin 1 is flipped, it lands on heads with probability .4; when coin 2 is flipped, it lands on heads with probability .7. One of these coins is randomly chosen and flipped 10 times.
(a) What is the probability that the coin lands on heads on exactly 7 of the 10 flips?
(b) Given that the first of these 10 flips lands heads, what is the conditional probability that exactly 7 of the 10 flips land on heads?
A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective. Find the expected number of defective items in the sample.
A box contains red and blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win ; if they are different colors, then you win . (That is, you lose .) Calculate
(a) the expected value of the amount you win;
(b) the variance of the amount you win.
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