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32 Pages

### problemset04

Course: STAT 416, Fall 2011
School: Purdue
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Word Count: 465

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416 August STAT/MA 29, 2011 Problem Set 4 Name: 1. Choosing a page at random. A student buys a brand new calculus textbook that has 1000 pages, numbered from 1 to 1000, of course. She randomly opens the book to a page and starts to read! Assume that any of the 1000 pages are equally likely to be chosen. Let B be the event that at least one of the digits on the chosen page is a 5 (for example, the page could be 572...

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416 August STAT/MA 29, 2011 Problem Set 4 Name: 1. Choosing a page at random. A student buys a brand new calculus textbook that has 1000 pages, numbered from 1 to 1000, of course. She randomly opens the book to a page and starts to read! Assume that any of the 1000 pages are equally likely to be chosen. Let B be the event that at least one of the digits on the chosen page is a 5 (for example, the page could be 572 or 505 or 355 or 115 etc., etc.). Let C be the event that at least two of the digits on the chosen page are 5 (for example, 355 or 575, etc.). (a.) Given that B occurs, nd the conditional probability of A, the event that 555 is the selected page. (b.) Given that C occurs, nd the conditional probability of A, the event that 555 is the selected page. 1 2. Socks. A matching pair of blue socks, a matching pair of red socks, and one lone white right-footed sock are in a drawer. Suppose that a person is looking for the two matching blue socks. He blindly pulls two socks out of the drawer simultaneously. Let B be the event that at least one of the socks is blue. (a.) Given that B occurs, nd the conditional probability of A, the event that both of the socks are blue. (b.) Given that B occurs, nd the conditional probability of Ac . 2 3. Seating arrangements. Alice, Catherine, Bob, Doug, and Edna are randomly assigned seats at a circular table in a perfectly circular room. Assume that rotations of the table do not matter, so there are exactly 24 possible outcomes in the sample space. Bob and Catherine are married. Doug and Edna are married. When people are married they love to sit beside each other. Let B denote the event that Bob and Catherine are sitting next to each other. Given that B occurs, nd the conditional probability of A, the event that Doug and Edna are sitting next to each other. 3 4. Pair of dice. Roll a pair of dice. Let B be the event that the two dice have dierent values. Given that B occurs, nd the conditional probability of A, the event that the sum of the dice is an even number. 4 5. Pair of dice. Roll a pair of dice. Let B be the event that the sum of the pair of dice is 9 or larger. Given that B occurs, nd the conditional probability of A, the event that the sum of the pair of dice is exactly 10. 5 6. Design your own problem and solution. Create your own problem about conditional probability. Design your problem in such a way that you would nd it enjoyable and also interesting (i.e., not completely trivial) if you found this problem in a probability book. Please provide the answer for your problem too. 6
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Purdue - STAT - 416
STAT/MA 416August 31, 2011Problem Set 5Name:1. Waking up at random.On each weekday, a student wakes up before 8 AM with probability .65, or after 8 AMwith probability .35.On each weekend, a student wakes up before 8 AM with probability .22, or afte
Purdue - STAT - 416
STAT/MA 416September 2, 2011Problem Set 6Name:1. Harmonicas. When ordering a new box of harmonicas, let X denote the time until thebox arrives, and let Y denote the number of harmonicas that work properly.Is X a continuous or discrete random variabl
Purdue - STAT - 416
STAT/MA 416September 7, 2011Problem Set 7Name:1. Butteries. Alice, Bob, and Charlotte are looking for butteries. They look in threeseparate parts of a eld, so that their probabilities of success do not aect each other. Alice nds 1 buttery with proba
Purdue - STAT - 416
STAT/MA 416September 9, 2011Problem Set 8Name:1. Butteries. Alice, Bob, and Charlotte are looking for butteries. They look in threeseparate parts of a eld, so that their probabilities of success do not aect each other. Alice nds 1 buttery with proba
Purdue - STAT - 416
STAT/MA 416September 12, 2011Problem Set 9Name:1. Butteries. Alice, Bob, and Charlotte are looking for butteries. They look in threeseparate parts of a eld, so that their probabilities of success do not aect each other. Alice nds 1 buttery with prob
Purdue - STAT - 416
STAT/MA 416September 14, 2011Problem Set 10Name:1. Butteries. Alice, Bob, and Charlotte are looking for butteries. They look in threeseparate parts of a eld, so that their probabilities of success do not aect each other. Alice nds 1 buttery with pro
Purdue - STAT - 416
STAT/MA 416September 16, 2011Problem Set 11Name:1a. Variance of an Indicator. Suppose event A occurs with probability p, and X is anindicator for A, i.e., X = 1 if A occurs, or X = 0 otherwise. We already know E(X ) = p.Find Var(X ).1b. Butteries.
Purdue - STAT - 416
STAT/MA 416September 19, 2011Problem Set 12Instructions for this assignment: Solve any 5 of the problems #1 through #8 that youchoose, and also solve problem #9. You can get some extra credit by solving the others too.Show all steps for all of your s
Purdue - STAT - 416
STAT/MA 416September 21, 2011Problem Set 13Name:1. Winnings and Losing. Suppose that a person wins a game of chance with probability0.40, and loses otherwise. If he wins, he earns 5 dollars, and if he loses, then he loses 4dollars.(a.) What is his
Purdue - STAT - 416
STAT/MA 416September 23, 2011Problem Set 14Name:1. Winnings and Losing. Suppose that a person wins a game of chance with probability0.40, and loses otherwise. If he wins, he earns 5 dollars, and if he loses, then he loses 4dollars. Assume that he pl
Purdue - STAT - 416
STAT/MA 416September 26, 2011Problem Set 15Name:1. Winnings and Losing. Suppose that a person wins a game of chance with probability0.40, and loses otherwise. If he wins, he earns 5 dollars, and if he loses, then he loses 4dollars. He plays the game
Purdue - STAT - 416
STAT/MA 416September 28, 2011Problem Set 16: By Request (from our dinner guests), a special Harry Potter homeworkName:1. Quidditch Training. Hermione is frustrated because she is extremely good at spellsbut she is struggling to learn how to y on her
Purdue - STAT - 416
STAT/MA 416September 30, 2011Problem Set 17Name:1. Hungry customers. At a certain hot dog stand, during the working day, the numberof people who arrive to eat is Poisson, with an average of 1 person every 2 minutes.a. What is the probability that ex
Purdue - STAT - 416
STAT/MA 416October 3, 2011Problem Set 18Name:1. Hungry customers. At a certain restaurant, during the working day, there are 12customers. Seven of them have pizza, and the other ve have burgers. Suppose that aperson is conducting a survey of three c
Purdue - STAT - 416
STAT/MA 416October 12, 2011Problem Set 19Name:1. Hearts. In a game of chance, you are allowed to shue a standard deck of cards andthen choose 3 cards randomly (without replacement). If two or more of them are hearts,then you win.a. What is the prob
Purdue - STAT - 416
STAT/MA 416October 14, 2011Problem Set 20Name:1. Rock Block. On a certain radio station, 70% of the songs are rock songs, and 30% ofthe songs are pop songs. The songs are selected independently. Each block of songs (ablock is a set of songs between
Purdue - STAT - 416
STAT/MA 416October 17, 2011Problem Set 21Name:1. Consider a random variable X with densityfX (x) =1 x/5e50for x &gt; 0,otherwise.a. Find P (3 X 5).b. Find an expression for the CDF FX (x) of X .c. Graph the CDF FX (x) of X .12. Let X have den
Purdue - STAT - 416
STAT/MA 416October 21, 2011Problem Set 23Name:1. Consider a pair of random variables X, Y with constant joint density on the triangle withvertices at (0, 0), (3, 0), and (0, 3).a. Are X and Y independent? Why or why not?b. Find the density fX (x) o
Purdue - STAT - 416
STAT/MA 416October 24, 2011Problem Set 24Name:1. Consider a pair of random variables X, Y with constant joint density on the triangle withvertices at (0, 0), (3, 0), and (0, 3).a. For 0 y 3, nd the conditional density fX |Y (x | y ) of X , given Y =
Purdue - STAT - 416
STAT/MA 416October 26, 2011Problem Set 25Name:1. Consider a pair of random variables X, Y with constant joint density on the triangle withvertices at (0, 0), (3, 0), and (0, 3).Find the expected value E(X ). (Notice that, by symmetry, E(Y ) is just
Purdue - STAT - 416
STAT/MA 416October 28, 2011Problem Set 26Name:1. Consider a pair of random variables X, Y with constant joint density on the triangle withvertices at (0, 0), (3, 0), and (0, 3).a. Find the expected value of the sum of X and Y , i.e., nd E(X + Y ).b
Purdue - STAT - 416
STAT/MA 416October 31, 2011Problem Set 27Name:1. Suppose that X has CDFFX (x) =0x241a. Find P (X 4.5).b. Find P (3.09 X 4.39).c. Find P (X 3.7).1x &lt; 2,2 x 6,x &gt; 6.2. Assume that the amount of soda in a can is uniformly distributed between
Purdue - STAT - 416
STAT/MA 416November 2, 2011Problem Set 28Name:1. The waiting time until the phone rings is exponential, with an average waiting time of30 minutes.a. What is the probability that nobody calls during the next hour?b. What is the standard deviation (i
Purdue - STAT - 416
STAT/MA 416November 4, 2011Problem Set 29Name:1. Let X1 , X2 , X3 be independent exponential waiting times, each with an average of 30minutes. Let Y = X1 + X2 + X3 .a. What is the average (in minutes) of Y ?b. What is the standard deviation (in min
Purdue - STAT - 416
STAT/MA 416November 7, 2011Problem Set 30Name:1. Assume X is normally distributed with E(X ) = 4.2 and Var X = 50.41.a. Find P (X 10).b. Find P (X 0).c. Find P (0 X 10).12. In a certain class, the student scores are approximately normally distrib
Purdue - STAT - 416
STAT/MA 416November 9, 2011Problem Set 31Name:1a. If X1 , X2 , X3 , X4 , X5 are independent random variables that each have average 8.2 andvariance 32.49, then the sum of the Xj s is normal. Thus, if we divide the sum of the Xj sby 5, we get the ave
Purdue - STAT - 416
STAT/MA 416November 11, 2011Problem Set 32Name:1. Each of the 30 students in a class each orders a package. They assume that the waitingtime (measured in weeks) for the packages are independent exponential random variables,with average waiting time
Purdue - STAT - 416
STAT/MA 416November 14, 2011Problem Set 33Name:1. There are 2000 ights that arrive at the Denver airport each day, 70% of which are ontime. What is the approximate probability that strictly more than 1420 of the ights are ontime in one day? Assume t
Purdue - STAT - 416
STAT/MA 416November 16, 2011Problem Set 34Name:1. Suppose that the number of Roseate Spoonbills (a very rare bird in Indiana) that yoverhead in 1 hour has a Poisson distribution with mean 2. Also suppose that the numberof Roseate Spoonbills is indep
Purdue - STAT - 416
STAT/MA 416November 28, 2011Problem Set 35Name:1. Jim cuts wood planks of length X for customers, where X is uniformly distributedbetween 10 and 14 feet. The price of a piece of wood is \$2 per foot, plus a at-rate surchargeof \$2 for Jims services. S
Purdue - STAT - 416
STAT/MA 416November 30, 2011Problem Set 36Name:1. Consider a tray with 8 lemonades and 3 raspberry juices. Alice and Bob each take 1drink from the tray, without replacement. Assume that all of their choices are equally likely.Let X be the number of
Purdue - STAT - 416
STAT/MA 416December 2, 2011Problem Set 37Name:1. Let X and Y have a joint uniform distribution on the triangle with corners at (0, 2),(2, 0), and the origin. Find E(Y | X = 1/2).12. Roll two 6-sided dice. Let X denote the minimum value that appears
Purdue - STAT - 416
STAT/MA 416December 5, 2011Problem Set 38Name:1a. The average amount of time that a student spends studying for a nal exam is 5 hours.Find an upper-bound on the probability that a student spends 7 or more hours studying fora nal exam.1b. Now also a
Purdue - STAT - 416
STAT/MA 416December 7, 2011Problem Set 39Name:1. Consider X1 , X2 , X3 , X4 which are independent and uniformly distributed on [0, 20].a. Find the density of the rst order statistic, i.e., nd fX(1) (x1 ).b. Find the density of the second order stati
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