MA2212 - HW06

MA2212 - HW06 - Department of Mathematics Name:

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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 3.2-1 MA2212 { . Y{ 3 Y 3 , zero elsewhere. HW06 Let the random variable X have the pdf X{Y{ (a) Sketch the graph of this pdf. (b) Determine and sketch the graph of the distribution function of X. { { . $ 3 3 (c) Find: (i) W{ 3 (ii) { { { 3I3 3 3 3 { { { { (iii) W{ {I (iv) W{ 4 { 4 3I3 { { { ' { " . ' { $' { . { . $ ' " % & $ ' $' # $ . { {Y{, where M(t) is the moment-generating function of a random variable of 3.2-12 Let W{Y{ the continuous type. Show that (a) W { { H { { { { H{ { H { { { { H { { H{ { W { { (b) # { ' . { . $ # ' # # Willis Thai { { MA2212 { H{ {H { { . {H { {{$ {H{ {{$ H{ {H { { . {H { {{$ {H{ {{$ HW06 { H { { . {H { {{$ $ Y {X 3.2-14 Find the moment-generating function M(t) of the distribution with pdf X{Y{ { {Y{ to determine the mean and the variance . Y . Use M(t) or W{Y{ {Y{ W{Y{ H{ { { { { { $ , { . { . { . { { { . {$ " { #" " {# #" { #" . { { 3.2-22 The weekly demand X for propane gas (in thousands of gallons) has the pdf X{Y{ Y X Y , Y . If two thousand gallons is in stock at the beginning of each week (and there is none extra during the week), what is the probability of not being able to meet the demand during a given week? {I 2 { 3.3-3 $ % . # Customers arrive randomly at a bank teller's window. Given that one customer arrived during a particular 10-minute period, let X equal the time within the 10 minutes that the customer {, find: arrived. If F is { (a) the pdf of X (b) W{ 4 { (c) W{ 3 {I 4 %{ { { #" { 3 I 3 %{ 3 { $ } % } % (d) E(X) I-I (e) Var(X) {I . I{$ $ - { . {$ Willis Thai 3.3-8 MA2212 HW06 Telephone calls enter a college switchboard according to a Poisson process on the average of two every 3 minutes. Let X denote the waiting time until the first call that arrives 10 AM. (a) What is the pdf of X? (b) Find W{ 2 { {I 4 { $ { { . } 3.3-13 Let X equal the number of bad records in each 100 feet of a used computer tape. Assume that X has a Poisson distribution with mean 2.5. Let W equal the number of feet before the first bad record is found. (a) Give the mean number of flaws per foot. (b) How is W distributed? Exponentially, where (c) Give the mean and variance of W. $ $ (d) Find: (i) W{ (ii) W{ (iii) W{ { { { 3 2 3 2 2 { { { $" # " &" # &" &" . . {{ } $" " } &" { { % { 2 2 2 { { "{ { $"{{ { $"{ "{ $"{ % % ( ( % % ( ( % % ...
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