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Make a histogram of the probability distribution. \\cribe what you see. ., S"

Find P(X < 3) and interpret the result. / "̂ "T \ ̂

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nd the probability distribution of T. 1~ ' 3 i- ^

ake a histogram of the probability distribution. 3 "^ ^ - &be what you see. f f s 7

nd P(T > 5) and interpret the result. : $ 5" = %

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Benford's law Faked numbers in tax returns, invoices,or expense account claims often display patterns thataren't present in legitimate records. Some patterns,like too many round numbers, are obvious and easilyavoided by a clever crook. Others are more subtle.It is a striking fact that the first digits of numbers inlegitimate records often follow a model known asBenford's law.5 Call the first digit of a randomly cho-sen record X for short. Benford's law gives this prob-ability model for X (note that a first digit can't be 0):

First digitX 1 2 3 4 5 6 7 8 9Probability: 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046

(a) Show that this is a legitimate probabilitydistribution.(b) Make a histogram of the probability distribution.Describe what you see.

(c) Describe the event X > 6 in words. What isP(X>6)?

(d) Express the event "first digit is at most 5" interms of X. What is the probability of this event?

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Benford's law Refer to Exercise 5. The first digit ofa randomly chosen expense account claim followsBenford's law. Consider the events A = first digit is 7or greater and B = first digit is odd.

(a) What outcomes make up the event A? What is

(b) What outcomes make up the event B? What isP(B)?(c) What outcomes make up the event "A or B"?What is P(A or B)? Why is this probability not equalto ?(A) + P(B)?

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Keno Keno is a favorite game in casinos, and similargames are popular with the states that operatelotteries. Balls numbered 1 -to 80 are tumbled in amachine as the bets are placed, then 20 of the ballsare chosen at random. Players select numbers bymarking a card. The simplest of the many wagersavailable is "Mark 1 Number." Your payoff is $3 ona $1 bet if the number you select is one of thosechosen. Because 20 of 80 numbers are chosen, yourprobability of winning is 20/80, or 0.25. LetX = theamount you gain on a single.play of the game.

(a) Make a table that shows the probability distribu-tion of X

(b) Compute the expected value of X. Explain whatthis result means for the player.

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19.) Housing in San Jose How do rented housing unitsdiffer from units occupied by their owners? Here arethe distributions of the number of rooms for owner-occupied units and renter-occupied units in San Jose,California:7

Number of Rooms

1 2 3 4 5 6 7 8 9 1 0

Owned 0.003 0.002 0.023 0.104 0.210 0.224 0.197 0.149 0.053 0.035

Rented 0.008 0.027 0.287 0.363 0.164 0.093 0.039 0.013 0.003 0.003

LetX = the number of rooms in a randomly selectedowner-occupied unit and Y = the number of roomsin a randomly chosen renter-occupied unit

(a) Make histograms suitable for comparing theprobability distributions of X and Y. Describe anydifferences that you observe.

(b) Find the mean number of rooms for both typesof housing unit. Explain why this difference makessense.

(c) Find the standard deviations of both X and Y.Explain why this difference makes sense.

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the owner distribution to have a slightly widerspread than the renter distribution. Eventhough the distribution of renter-occupiedunits is skewed to the right, it is more concen-trated (contains less variability) about the"peak" than the symmetric distribution.for.,owner-occupied units.

I 23| ITBS scores The Normal distribution with mean., , fj, = 6.8 and standard deviation a — 1.6 is a good

description of the Iowa Test of Basic Skills (ITBS)vocabulary scores of seventh-grade students in Gary,Indiana. Call the score of a randomly chosen studentX for short. Find ?(X > 9) and interpret the result.Follow the four-step process.

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