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(See Exercises 3 and 4 from "Problems on Random Variables and Probabilities"). The class { C j : 1 j 10 } is a partition. Random variable X has values { 1 , 3 , 2 , 3 , 4 , 2 , 1 , 3 , 5 , 2 } on C 1 through C 10 , respectively, with probabilities 0.08, 0.13, 0.06, 0.09, 0.14, 0.11, 0.12, 0.07, 0.11, 0.09.Determine and plot the distribution function F X .

T = [1 3 2 3 4 2 1 3 5 2];pc = 0.01*[8 13 6 9 14 11 12 7 11 9];[X,PX] = csort(T,pc);ddbn Enter row matrix of VALUES XEnter row matrix of PROBABILITIES PX % See MATLAB plot
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(See Exercise 6 from "Problems on Random Variables and Probabilities"). A store has eight items for sale. The prices are $3.50, $5.00, $3.50, $7.50, $5.00, $5.00, $3.50, and $7.50, respectively.A customer comes in. She purchases one of the items with probabilities 0.10, 0.15, 0.15, 0.20, 0.10 0.05, 0.10 0.15. Therandom variable expressing the amount of her purchase may be written

X = 3 . 5 I C 1 + 5 . 0 I C 2 + 3 . 5 I C 3 + 7 . 5 I C 4 + 5 . 0 I C 5 + 5 . 0 I C 6 + 3 . 5 I C 7 + 7 . 5 I C 8

Determine and plot the distribution function for X .

T = [3.5 5 3.5 7.5 5 5 3.5 7.5];pc = 0.01*[10 15 15 20 10 5 10 15];[X,PX] = csort(T,pc);ddbn Enter row matrix of VALUES XEnter row matrix of PROBABILITIES PX % See MATLAB plot
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(See Exercise 12 from "Problems on Random Variables and Probabilities"). The class { A , B , C , D } has minterm probabilities

p m = 0 . 001 * [ 5 7 6 8 9 14 22 33 21 32 50 75 86 129 201 302 ]

Determine and plot the distribution function for the random variable X = I A + I B + I C + I D , which counts the number of the events which occur on a trial.

npr06_12 Minterm probabilities in pm, coefficients in c T = sum(mintable(4)); % Alternate solution. See Exercise 12 from "Problems on Random Variables and Probabilities" [X,PX]= csort(T,pm); ddbnEnter row matrix of VALUES X Enter row matrix of PROBABILITIES PX % See MATLAB plot
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Suppose a is a ten digit number. A wheel turns up the digits 0 through 9 with equal probability on each spin. On ten spins what is the probabilityof matching, in order, k or more of the ten digits in a , 0 k 10 ? Assume the initial digit may be zero.

P = c b i n o m ( 10 , 0 . 1 , 0 : 10 ) .

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In a thunderstorm in a national park there are 127 lightning strikes. Experience shows that the probability of of a lightning strike starting a fire is about0.0083. What is the probability that k fires are started, k = 0 , 1 , 2 , 3 ?

P = ibinom(127,0.0083,0:3) P = 0.3470 0.3688 0.1945 0.0678

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A manufacturing plant has 350 special lamps on its production lines. On any day, each lamp could fail with probability p = 0 . 0017 . These lamps are critical, and must be replaced as quickly as possible. It takes about one hour toreplace a lamp, once it has failed. What is the probability that on any day the loss of production time due to lamp failaures is k or fewer hours, k = 0 , 1 , 2 , 3 , 4 , 5 ?

P = 1 - cbinom(350,0.0017,1:6)

= 0.5513 0.8799 0.9775 0.9968 0.9996 1.0000
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Two hundred persons buy tickets for a drawing. Each ticket has probability 0.008 of winning. What is the probability of k or fewer winners, k = 2 , 3 , 4 ?

P = 1 - cbinom(200,0.008,3:5) = 0.7838 0.9220 0.9768

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Two coins are flipped twenty times. What is the probability the results match (both heads or both tails) k times, 0 k 20 ?

P = ibinom(20,1/2,0:20)

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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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