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Chapter 4: Q. 4.33 (page 172)
Repeat Theoretical Exercise 4.32, this time assuming that withdrawn chips are not replaced before the next selection.
Probability mass function does not exist
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A fair coin is continually flipped until heads appears for the 10th time. Let X denote the number of tails that occur. Compute the probability mass function of X.
A communications channel transmits the digits and However, due to static, the digit transmitted is incorrectly received with probability Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit instead of and 11111 instead of If the receiver of the message uses 鈥渕ajority鈥� decoding, what is the probability that the message will be wrong when decoded? What independence assumptions are you making?
Show how the derivation of the binomial probabilities leads to a proof of the binomial theorem when and are nonnegative.
Hint: Let .
A student is getting ready to take an important oral examination and is concerned about the possibility of having an 鈥渙n鈥� day or an 鈥渙ff鈥� day. He figures that if he has an on the day, then each of his examiners will pass him, independently of one another, with probability, whereas if he has an off day, this probability will be reduced to. Suppose that the student will pass the examination if a majority of the examiners pass him. If the student believes that he is twice as likely to have an off day as he is to have an on the day, should he request an examination withexaminers or withexaminers?
Suppose that the distribution function of X given by
(a) Find .
(b) Find .
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