大家帮我看看这几道题,谢谢了..急
<p>1.<br/>.A television store owner figures that 50% of the customers entering his store will<br/>purchase an ordinary television set, 20% will purchase a high definition television set,<br/>and 30% will just be browsing. If five customers enter the store on a certain day, what is<br/>the probability that two customers purchase an HDTV, one customer purchases a color se<br/>t, and two customers purchase nothing?<br/><br/>2. Suppose that two teams are playing a series of games, each of which is independently<br/>won by team A with probability p and by team B with probability 1-p. The winner of the<br/>series is the first team to win k games.<br/>a) If k=4, what is the probability that a total of 7 games are played? Show that this<br/>probability is maximized when k=1/2.<br/>b) Find the expected number of games that are played when k=2; when k=3; and when<br/>k=4.<br/></p><p>3.</p><p>Suppose that each coupon obtained is, independent of what has been previously obtained,<br/>equally likely to be any of m different types. Find the expected number of coupons one<br/>needs to obtain in order to have at least one of each type.<br/>Hint: Let X be the number needed. It is useful to represent X by</p><p> m</p><p>X=Σ Xi , where each Xi is a geometric random variable.</p><p> i=1</p><p>(这一题的最后X的表达式我没有用论坛的数学表达式方法输入,但是大家看得懂吧)</p><p></p><p>4<br/>A coin, having probability p of landing heads, is flipped until head appears for the r th<br/>time. Let N denote the number of flips required. Calculate E.<br/>Hint: There is an easy way of doing this – it involves writing N as the sum of r geometric<br/>random variables.</p><p>5<br/>Let E and F be mutually exclusive events in the sample space of an experiment.<br/>Suppose that the experiment is repeated until either event E or event F occurs. What<br/>does the sample space of this new “super” experiment look like? Show that the<br/>probability that the event E occurs before the event F is P(E)/{P(E)+P(F)}<br/>Hint: Argue that the probability that the original experiment is performed n times and E<br/> n-1</p><p>occurs on the nth time is P(E)*(1-p) , n=1,2,…, where p = P(E)+P(F). Add these<br/>probabilities to get the desired answer. ( 这里n-1是(1-p)的n-1次方,大家注意.我同样写的不是很清楚.)</p><p>6<br/>Suppose that we want to generate a random variable X that is equally likely to be either 0<br/>or 1, and all that we have at our disposal is a biased coin that, when flipped, lands on<br/>heads with some (unknown) probability p. Consider the following procedure:<br/>1. Flip the coin, and let O1, either heads or tails be the result.<br/>2. Flip the coin, and let O2 be the result.<br/>3. If O1 and O2 are the same, return to step 1.<br/>4. If O2 is heads, set X=0 and, otherwise set X=1.<br/>Show that the random variable X generated by this procedure is equally like to be either 0<br/>or 1.</p><p></p> 你先把它翻译后，再弄上去。
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