1、機(jī)率midterm, Fall 2004姓名: _ 學(xué)號(hào): _ 分?jǐn)?shù): _1. Let be a Poisson() random variable.Find 2. Let be a Binomial(, ) random variable. Show that , , .3. An absentminded professor does not remember which of his 12 keys will open his office door. If he tries them at random and with replacement:(a) On average, how

2、many keys should he try before his door opens?(b) What is the probability that he opens his office door after only 3 tries?(c) Find 4. Ann puts at most one piece of fruit in her childs lunch bag everyday. If she has only 3 oranges and 2 apples for the next 8 lunches of her child, in how many ways ca

3、n she do this?4. Of the 28 professors in a certain department, 18 drive foreign and 10 drive domestic cars. If 5 of these professors are selected at random, let equal the number of them drive foreign car. Find 5. Suppose that there was a cancer diagnostic test that was 95% accurate both on those tha

4、t do and those that do not have the disease. (已知實(shí)際有癌癥, 檢定出來(lái)是有癌癥的機(jī)率為0.95, 已知實(shí)際沒(méi)有癌癥, 檢定出來(lái)是沒(méi)有癌癥的機(jī)率為0.95) If 0.004=0.4% of the population have cancer, compute the probability that a test person has cancer, given that his or her test result indicate so.(求已知檢定出來(lái)是有癌癥, 此人實(shí)際有癌癥的機(jī)率)6. A box has 8 red and 10 b

5、lack balls. A ball is selected from the box. If the ball is black, it is returned to the box. If the ball is red, it and 3 additional red balls are added to the box. Find the probability that a second ball selected from the box is red.7. The suicide rate in a certain state in 1 suicide per 100000 in

6、habitants per month. (在某州自殺率為每個(gè)月, 每十萬(wàn)居民有一個(gè))(a) Find the probability that in a city of 400000 inhabitants within this state, there will be 2 or more suicides in a given month. (在此州某城市有四十萬(wàn)居民, 請(qǐng)問(wèn)在某個(gè)月至少有2次或多於2次自殺的機(jī)率)(b) What is the probability that there will be at least 3 months during the year that wi

7、ll have 2 or more suicides?(求在一年12月中，至少有3個(gè)月會(huì)有2次或多於2次自殺的機(jī)率)8. is the cdf of (a) Sketch the graph of .畫(huà)在函數(shù)旁(b) Compute 9. Let the pmf be positive at , and zero elsewhere.(a) If , find .(b) If and if , find and .10. Suppose is a random variable with the following pmf-3-2023(a) Find the value of .(b) Fi

8、nd .(c) Find .11. , are events with , , , and are disjoint, and are independent, and . Find 12. Figure 3.4 shows an electric circuit (電路) in which each of the switches located at 1,2,3, and 4 is independently closed or open with probabilities and , respectively. If a signal is fed to the input, what

9、 is the probability that it transmitted to the output?13. Suppose is a discrete random variable with and , Find .14. Sharon and Ann play a series of backgammon (西洋雙陸棋) games until one of them win 5 games. Suppose that the games are independent and the probability that Sharon wins a game is 0.58. Fin

10、d the probability the series ends in 7 games.15. If 4 Americans, 3 Frenchmen, and 3 Englishmen are to be seated in a row, how many seating arrangements are possible when people of the same nationality must sit next to each other?16. Find the coefficient of in the expansion of .17. 10 identical erasers are to be divided