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Unformatted text preview: Department of Mathematics Name: ________________________________ Poly ID: _______________________________ Instructor's Name: _______________________ Course No:_____________________________ Course Section: _________________________ Homework Section No: ___________________ Willis Thai MA2212 HW03 2.12 Let a chip be taken at random from a bowl that contains six white chips,t here red chips, and one blue chip. Let the random variable X = 1 if the outcome is a white chip; let X = 5 if the outcome is a red chip; and let X = 10 if the outcome is a blue chip. (a) Find the p.m.f. of X. {I {I { { (b) Graph the p.m.f. as a bar graph. {I { 2.110 Suppose there are 3 defective items in a lot of 50 items. A sample size of 10 is taken at random and without replacement. Let X denote the number of defective items in the sample. Find the probability that the sample contains (a) Exactly 1 defective item. % % %% % (b) At most 1 defective item. 2.111 In a lot of 100 light bulbs, there are 5 bad bulbs. An inspector inspects 10 bulbs selected at random. Find the probability of finding at least one defective bulb. HINT: First compute the probability of finding no defectives in the sample. % . 2.113 Let X be the number of accidents in a factory per week having p.m.f. Willis Thai X{Y{ MA2212 Y HW03 {Y {{Y Find the conditional probability of { { (&  .  F  .  F  {. {I 4 I4 { {I 4 { 4 , given that { {I 4 I 4 { {I 4 { {I 4 { { .  F J J
# ' # $ 4 . HINT: Write X{Y{ Y . Y . 2.114 Often in buying a product at a supermarket, there is a concern about that item being underweight. Suppose there are 20 "onepound" packages of frozen ground turkey on display and 3 of them are underweight. A consumer group buys five of the 20 packages at random. What is the probability of at least one of the five being underweight? {I 4 { 2.21 . {I { . % Let the random variable X be the number of days that a certain patient needs to be in the hospital. Say X had the p.m.f. X{Y{
Y , Y If the patient is to receive from an insurance company $200 for each for the first two days in the hospital and $100 for each day after the first two days, what is the expected payment for the hospitalization? { { { { { { { { { { { { { { { { { { { { F{ { F{ { F{ { F 2.22 An insurance company sells an automobile policy with a deductible of one unit. Let X be the amount of the loss having p.m.f. X{Y{ Y Y Y where c is a constant. Determine c and the expected value of the amount the insurance company must pay. %I { { %      F 0 %  0  0  0  0  0 F I % % Willis Thai MA2212 HW03 2.25 In a particular lottery 3,000,000 tickets are sold each week for 50 apiece. Out of the 3,000,000 tickets, 12,006 are drawn at random and without replacement and awarded prizes: twelve thousand $25 prizes, four $10,000 prizes, one $50,000 prize, and one $200,000 prize. If you purchased a single ticket each week, what is the expected value of this game to you? %% %% { { { {. { F{ { F{ F { {
# %"""""" { { # %"""""" . 2.214 Suppose that a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students for a total of 1000 students. (a) What is the average class size? 0  0  0 (b) Select a student randomly out of the 1000 students. Let the random variable X equal the size of the class to which this student belongs and define the p.m.f. of X. {I {I {I { { (c) Find E(X), the expected value of X. Does this answer surprise you? {I{ 2.31 (a) X{Y{
$ { { { F{ { Y F{ { F . Find the mean and variance for the following discrete distributions: , F{ F{ {{$ {F{ {F{ {F{ { { . {. {$ (c) X{Y{
$ (b) X{Y{ , { {{ { $ { . {{ {
Y F  {. {$ Y F{ . . F  { {$ {$ F{ F  { {$ . {$ F{ F{ {$ . F {$ F{ . {$ F , F{ {. F $ F{ { F Y . F{ {
$ $ . F F . F F % Willis Thai MA2212 HW03 2.315 A fair coin is flipped successively at random until the first head is observed. Let the random variable X denote the number of flips of the coin that are required. Then the space of X is {. Later we learn that under certain conditions, we can assign W {Y Y probabilities to these outcomes in S with the function X{Y{ negative binomial { . { , with F $ Y Y . Compute and take the mean . HINT: Write out the series for . and then construct the series for the difference. An alternative method would be to compare the series for F % $ to that of the . F  # F  J F # $  {J . { F % F .J F J F # F # . . .J F 2.316 Let X equal the number of calls per hour received by 911 between midnight and noon and reported in the Holland Sentinel. On October 29 and October 30, the number of the following numbers of call were reported: 0 1 1 1 0 1 2 1 4 1 2 3 0 3 0 1 0 1 1 2 3 0 2 2 (a) Find the sample mean. % { { F{ { F{ {
$ $ (b) Find the sample variance. %% % . F
$ F{ { F F{ { . F
$ F . F
$ F . F $ F F . F $ F 2.319 A warranty is written on a product worth $10,000 so that the buyer is given $8000 if it fails in the first year, $6000 if it fails in the second, $4000 if it fails in the third, $2000 if it fails in the fourth, and zero after that. Its probability of failing in a year is 0.1: failures are independent of those of the other years. What is the expected value of the warranty? { { {% {{ {  { {{ %{{ {  { {{ %{$ { {  { {{ %{% { { { { % %% ...
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 Fall '08
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