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Chapter 4: Q.4.25 (page 165)
Two coins are to be 铿俰pped. The 铿乺st coin will land on heads with probability ., the second with probability .. Assume that the results of the 铿俰ps are independent, and let X equal the total number of heads that result. (a) Find P{X =}. (b) Determine E[X].
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If X has distribution function F, what is the distribution function of the random variable 伪X + 尾, where 伪 and 尾 are constants,
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?
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?
There are types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of typewith probability , . If n coupons are collected, find the expected number of distinct types that appear in this set. (That is, find the expected number of types of coupons that appear at least once in the set of coupons.)
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 Two marbles are withdrawn randomly.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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