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Statistics LibreTexts

15.3: Problems on Random Selection

  • Page ID
    10846
    • Contributed by Paul Pfeiffer
    • Professor emeritus (Computational and Applied Mathematics) at Rice University

    Exercise \(\PageIndex{1}\)

    (See Exercise 3 from "Problems on Random Variables and Joint Distributions") A die is rolled. Let \(X\) be the number of spots that turn up. A coin is flipped \(X\) times. Let \(Y\) be the number of heads that turn up. Determine the distribution for \(Y\).

    Answer
    PX = [0 (1/6)*ones(1,6)];
    PY = [0.5 0.5];
    gend
    Do not forget zero coefficients for missing powers
    Enter gen fn COEFFICIENTS for gN  PX
    Enter gen fn COEFFICIENTS for gY  PY
    Results are in N, PN, Y, PY, D, PD, P
    May use jcalc or jcalcf on N, D, P
    To view the distribution, call for gD.
    disp(gD)             % Compare with P8-3
             0    0.1641
        1.0000    0.3125
        2.0000    0.2578
        3.0000    0.1667
        4.0000    0.0755
        5.0000    0.0208
        6.0000    0.0026

    Exercise \(\PageIndex{2}\)

    (See Exercise 4 from "Problems on Random Variables and Joint Distributions") As a variation of Exercise 15.3.1, suppose a pair of dice is rolled instead of a single die. Determine the distribution for \(Y\).

    Answer
    PN = (1/36)*[0 0 1 2 3 4 5 6 5 4 3 2 1];
    PY = [0.5 0.5];
    gend
    Do not forget zero coefficients for missing powers
    Enter gen fn COEFFICIENTS for gN  PN
    Enter gen fn COEFFICIENTS for gY  PY
    Results are in N, PN, Y, PY, D, PD, P
    May use jcalc or jcalcf on N, D, P
    To view the distribution, call for gD.
    disp(gD)
             0    0.0269
        1.0000    0.1025
        2.0000    0.1823
        3.0000    0.2158
        4.0000    0.1954
        5.0000    0.1400
        6.0000    0.0806
        7.0000    0.0375
        8.0000    0.0140     % (Continued next page)
        9.0000    0.0040
       10.0000    0.0008
       11.0000    0.0001
       12.0000    0.0000

    Exercise \(\PageIndex{3}\)

    (See Exercise 5 from "Problems on Random Variables and Joint Distributions") Suppose a pair of dice is rolled. Let \(X\) be the total number of spots which turn up. Roll the pair an additional \(X\) times. Let \(Y\) be the number of sevens that are thrown on the \(X\) rolls. Determine the distribution for \(Y\). What is the probability of three or more sevens?

    Answer
    PX = (1/36)*[0 0 1 2 3 4 5 6 5 4 3 2 1];
    PY = [5/6 1/6];
    gend
    Do not forget zero coefficients for missing powers
    Enter gen fn COEFFICIENTS for gN  PX
    Enter gen fn COEFFICIENTS for gY  PY
    Results are in N, PN, Y, PY, D, PD, P
    May use jcalc or jcalcf on N, D, P
    To view the distribution, call for gD.
    disp(gD)
             0    0.3072
        1.0000    0.3660
        2.0000    0.2152
        3.0000    0.0828
        4.0000    0.0230
        5.0000    0.0048
        6.0000    0.0008
        7.0000    0.0001
        8.0000    0.0000
        9.0000    0.0000
       10.0000    0.0000
       11.0000    0.0000
       12.0000    0.0000
       P = (D>=3)*PD'
    P =  0.1116

    Exercise \(\PageIndex{4}\)

    (See Example 7 from "Conditional Expectation, Regression") A number \(X\) is chosen by a random selection from the integers 1 through 20 (say by drawing a card from a box). A pair of dice is thrown \(X\) times. Let \(Y\) be the number of “matches” (i.e., both ones, both twos, etc.). Determine the distribution for \(Y\).

    Answer
    gN = (1/20)*[0 ones(1,20)];
    gY = [5/6 1/6];
    gend
    Do not forget zero coefficients for missing powers
    Enter gen fn COEFFICIENTS for gN  gN
    Enter gen fn COEFFICIENTS for gY  gY
    Results are in N, PN, Y, PY, D, PD, P
    May use jcalc or jcalcf on N, D, P
    To view the distribution, call for gD.
     
    disp(gD)
             0    0.2435
        1.0000    0.2661
        2.0000    0.2113
        3.0000    0.1419
        4.0000    0.0795
        5.0000    0.0370
        6.0000    0.0144
        7.0000    0.0047
        8.0000    0.0013
        9.0000    0.0003
       10.0000    0.0001
       11.0000    0.0000
       12.0000    0.0000
       13.0000    0.0000
       14.0000    0.0000
       15.0000    0.0000
       16.0000    0.0000
       17.0000    0.0000
       18.0000    0.0000
       19.0000    0.0000
       20.0000    0.0000

    Exercise \(\PageIndex{5}\)

    (See Exercise 20 from "Problems on Conditional Expectation, Regression") A number \(X\) is selected randomly from the integers 1 through 100. A pair of dice is thrown \(X\) times. Let \(Y\) be the number of sevens thrown on the \(X\) tosses. Determine the distribution for \(Y\). Determine \(E[Y]\) and \(P(Y \le 20)\).

    Answer
    gN = 0.01*[0 ones(1,100)];
    gY = [5/6 1/6];
    gend
    Do not forget zero coefficients for missing powers
    Enter gen fn COEFFICIENTS for gN  gN
    Enter gen fn COEFFICIENTS for gY  gY
    Results are in N, PN, Y, PY, D, PD, P
    May use jcalc or jcalcf on N, D, P
    To view the distribution, call for gD.
    EY = dot(D,PD)
    EY =   8.4167
    P20 = (D<=20)*PD'
    P20 =  0.9837

    Exercise \(\PageIndex{6}\)

    (See Exercise 21 from "Problems on Conditional Expectation, Regression") A number \(X\) is selected randomly from the integers 1 through 100. Each of two people draw \(X\) times independently and randomly a number from 1 to 10. Let \(Y\) be the number of matches (i.e., both draw ones, both draw twos, etc.). Determine the distribution for \(Y\). Determine \(E[Y]\) and \(P(Y \le 10)\).

    Answer
    gN = 0.01*[0 ones(1,100)];
    gY = [0.9 0.1];
    gend
    Do not forget zero coefficients for missing powers
    Enter gen fn COEFFICIENTS for gN  gN
    Enter gen fn COEFFICIENTS for gY  gY
    Results are in N, PN, Y, PY, D, PD, P
    May use jcalc or jcalcf on N, D, P
    To view the distribution, call for gD.
    EY = dot(D,PD)
    EY =  5.0500
    P10 = (D<=10)*PD'
    P10 = 0.9188

    Exercise \(\PageIndex{7}\)

    Suppose the number of entries in a contest is \(N\) ~ binomial (20, 0.4). There are four questions. Let \(Y_i\) be the number of questions answered correctly by the \(i\)th contestant. Suppose the \(Y_i\) are iid, with common distribution

    \(Y =\) [1 2 3 4] \(PY =\) [0.2 0.4 0.3 0.1]

    Let \(D\) be the total number of correct answers. Determine \(E[D]\), \(\text{Var} [D]\), \(P(15 \le D \le 25)\), and \(P(10 \le D \le 30)\).

    Answer
    gN = ibinom(20,0.4,0:20);
    gY = 0.1*[0 2 4 3 1];
    gend
    Do not forget zero coefficients for missing powers
    Enter gen fn COEFFICIENTS for gN  gN
    Enter gen fn COEFFICIENTS for gY  gY
    Results are in N, PN, Y, PY, D, PD, P
    May use jcalc or jcalcf on N, D, P
    To view the distribution, call for gD.
    ED = dot(D,PD)
    ED =  18.4000
    VD = (D.^2)*PD' - ED^2
    VD =  31.8720
    P1 = ((15<=D)&(D<=25))*PD'
    P1 =   0.6386
    P2 = ((10<=D)&(D<=30))*PD'
    P2 =   0.9290

    Exercise \(\PageIndex{8}\)

    Game wardens are making an aerial survey of the number of deer in a park. The number of herds to be sighted is assumed to be a random variable \(N\) ~ binomial (20, 0.5). Each herd is assumed to be from 1 to 10 in size, with probabilities

    Value 1 2 3 4 5 6 7 8 9 10
    Probability 0.05 0.10 0.15 0.20 0.15 0.10 0.10 0.05 0.05 0.05

    Let \(D\) be the number of deer sighted under this model. Determine \(P(D \le t)\) for \(t = 25, 50, 75, 100\) and \(P(D \ge 90)\).

    Answer
    gN = ibinom(20,0.5,0:20);
    gY = 0.01*[0 5 10 15 20 15 10 10 5 5 5];
    gend
    Do not forget zero coefficients for missing powers
    Enter gen fn COEFFICIENTS for gN  gN
    Enter gen fn COEFFICIENTS for gY  gY
    Results are in N, PN, Y, PY, D, PD, P
    May use jcalc or jcalcf on N, D, P
    To view the distribution, call for gD.
    k = [25 50 75 100];
    P = zeros(1,4);
    for i = 1:4
        P(i) = (D<=k(i))*PD';
    end
    disp(P)
        0.0310    0.5578    0.9725    0.9998

    Exercise \(\PageIndex{9}\)

    A supply house stocks seven popular items. The table below shows the values of the items and the probability of each being selected by a customer.

    Value 12.50 25.00 30.50 40.00 42.50 50.00 60.00
    Probability 0.10 0.15 0.20 0.20 0.15 0.10 0.10

    Suppose the purchases of customers are iid, and the number of customers in a day is binomial (10,0.5). Determine the distribution for the total demand \(D\).

    1. How many different possible values are there? What is the maximum possible total sales?
    2. Determine \(E[D]\) and \(P(D \le t)\) for \(t = 100, 150, 200, 250, 300\).
      Determine \(P(100 < D \le 200)\).
    Answer
    gN = ibinom(10,0.5,0:10);
    Y  = [12.5 25 30.5 40 42.5 50 60];
    PY = 0.01*[10 15 20 20 15 10 10];
    mgd
    Enter gen fn COEFFICIENTS for gN  gN
    Enter VALUES for Y  Y
    Enter PROBABILITIES for Y  PY
    Values are in row matrix D; probabilities are in PD.
    To view the distribution, call for mD.
    s = size(D)
    s =    1   839
    M = max(D)
    M =    590
    t = [100 150 200 250 300];
    P = zeros(1,5);
    for i = 1:5
        P(i) = (D<=t(i))*PD';
    end
    disp(P)
        0.1012    0.3184    0.6156    0.8497    0.9614
    P1 = ((100<D)&(D<=200))*PD'
    P1 =   0.5144

    Exercise \(\PageIndex{10}\)

    A game is played as follows:

    1. A wheel is spun, giving one of the integers 0 through 9 on an equally likely basis.
    2. A single die is thrown the number of times indicated by the result of the spin of the wheel. The number of points made is the total of the numbers turned up on the sequence of throws of the die.
    3. A player pays sixteen dollars to play; a dollar is returned for each point made.

    Let \(Y\) represent the number of points made and \(X = Y - 16\) be the net gain (possibly negative) of the player. Determine the maximum value of

    \(X, E[X], \text{Var} [X], P(X > 0), P(X \ge 10), P(X \ge 16)\)

    Answer
    gn = 0.1*ones(1,10);
    gy = (1/6)*[0 ones(1,6)];
    [Y,PY] = gendf(gn,gy);
    [X,PX] = csort(Y-16,PY);
    M = max(X)
    M =  38
    EX = dot(X,PX)               % Check EX = En*Ey - 16 = 4.5*3.5
    EX  =  -0.2500               % 4.5*3.5 - 16 = -0.25
    VX = dot(X.^2,PX) - EX^2
    VX =  114.1875
    Ppos = (X>0)*PX'
    Ppos =  0.4667
    P10 = (X>=10)*PX'
    P10 =   0.2147
    P16 = (X>=16)*PX'
    P16 =   0.0803

    Exercise \(\PageIndex{11}\)

    Marvin calls on four customers. With probability \(p_1 = 0.6\) he makes a sale in each case. Geraldine calls on five customers, with probability \(p_2 = 0.5\) of a sale in each case. Customers who buy do so on an iid basis, and order an amount \(Y_i\) (in dollars) with common distribution:

    \(Y =\) [200 220 240 260 280 300] \(PY =\) [0.10 0.15 0.25 0.25 0.15 0.10]

    Let \(D_1\) be the total sales for Marvin and \(D_2\) the total sales for Geraldine. Let \(D = D_1 + D_2\). Determine the distribution and mean and variance for \(D_1\), \(D_2\), and \(D\). Determine \(P(D_1 \ge D_2)\) and \(P(D \ge 1500)\), \(P(D \ge 1000)\), and \(P(D \ge 750)\).

    Answer
    gnM = ibinom(4,0.6,0:4);
    gnG = ibinom(5,0.5,0:5);
    Y = 200:20:300;
    PY = 0.01*[10 15 25 25 15 10];
    [D1,PD1] = mgdf(gnM,Y,PY);
    [D2,PD2] = mgdf(gnG,Y,PY);
    ED1 = dot(D1,PD1)
    ED1 =  600.0000              % Check: ED1 = EnM*EY = 2.4*250
    VD1 = dot(D1.^2,PD1) - ED1^2
    VD1 =    6.1968e+04
    ED2 = dot(D2,PD2)
    ED2 =  625.0000              % Check: ED2 = EnG*EY = 2.5*250
    VD2 = dot(D2.^2,PD2) - ED2^2
    VD2 =    8.0175e+04
    [D1,D2,t,u,PD1,PD2,P] = icalcf(D1,D2,PD1,PD2);
    Use array opertions on matrices X, Y, PX, PY, t, u, and P
    [D,PD] = csort(t+u,P);
    ED = dot(D,PD)
    ED =   1.2250e+03
    eD = ED1 + ED2              % Check: ED = ED1 + ED2
    eD =   1.2250e+03           % (Continued next page)
     
    VD = dot(D.^2,PD) - ED^2
    VD =   1.4214e+05
    vD = VD1 + VD2            % Check: VD = VD1 + VD2
    vD =   1.4214e+05
    P1g2 = total((t>u).*P)
    P1g2 = 0.4612
    k = [1500 1000 750];
    PDk = zeros(1,3);
    for i = 1:3
       PDk(i) = (D>=k(i))*PD';
    end
    disp(PDk)
        0.2556    0.7326    0.8872

    Exercise \(\PageIndex{12}\)

    A questionnaire is sent to twenty persons. The number who reply is a random number \(N\) ~ binomial (20, 0.7). If each respondent has probability \(p = 0.8\) of favoring a certain proposition, what is the probability of ten or more favorable replies? Of fifteen or more?

    Answer
    gN = ibinom(20,0.7,0:20);
    gY = [0.2 0.8];
    gend
    Do not forget zero coefficients for missing powers
    Enter gen fn COEFFICIENTS for gN  gN
    Enter gen fn COEFFICIENTS for gY  gY
    Results are in N, PN, Y, PY, D, PD, P
    May use jcalc or jcalcf on N, D, P
    To view the distribution, call for gD.
    P10 = (D>=10)*PD'
    P10 =   0.7788
    P15 = (D>=15)*PD'
    P15 =   0.0660
    pD = ibinom(20,0.7*0.8,0:20);  % Alternate: use D binomial (pp0)
    D = 0:20;
    p10 = (D>=10)*pD'
    p10 =  0.7788
    p15 = (D>=15)*pD'
    p15 =  0.0660

    Exercise \(\PageIndex{13}\)

    A random number \(N\) of students take a qualifying exam. A grade of 70 or more earns a pass. Suppose \(N\) ~ binomial (20, 0.3). If each student has probability \(p = 0.7\) of making 70 or more, what is the probability all will pass? Ten or more will pass?

    Answer
    gN = ibinom(20,0.3,0:20);
    gY = [0.3 0.7];
    gend
    Do not forget zero coefficients for missing powers
    Enter gen fn COEFFICIENTS for gN  gN
    Enter gen fn COEFFICIENTS for gY  gY
    Results are in N, PN, Y, PY, D, PD, P
    May use jcalc or jcalcf on N, D, P
    To view the distribution, call for gD.
    Pall = (D==20)*PD'
    Pall =  2.7822e-14
    pall = (0.3*0.7)^20    % Alternate: use D binomial (pp0)
    pall =  2.7822e-14
    P10 = (D >= 10)*PD'
    P10 = 0.0038

    Exercise \(\PageIndex{14}\)

    Five hundred questionnaires are sent out. The probability of a reply is 0.6. The probability that a reply will be favorable is 0.75. What is the probability of at least 200, 225, 250 favorable replies?

    Answer
    n = 500;
    p = 0.6;
    p0 = 0.75;
    D = 0:500;
    PD = ibinom(500,p*p0,D);
    k = [200 225 250];
    P = zeros(1,3);
    for i = 1:3
       P(i) = (D>=k(i))*PD';
    end
    disp(P)
        0.9893    0.5173    0.0140

    Exercise \(\PageIndex{15}\)

    Suppose the number of Japanese visitors to Florida in a week is \(N1\) ~ Poisson (500) and the number of German visitors is \(N2\) ~ Poisson (300). If 25 percent of the Japanese and 20 percent of the Germans visit Disney World, what is the distribution for the total number \(D\) of German and Japanese visitors to the park? Determine \(P(D \ge k)\) for \(k = 150, 155, \cdot\cdot\cdot, 245, 250\).

    Answer

    \(JD\) ~ Poisson (500*0.25 = 125); \(GD\) ~ Poisson (300*0.20 = 60); \(D\) ~ Poisson (185).

    k = 150:5:250;
    PD = cpoisson(185,k);
    disp([k;PD]')
      150.0000    0.9964
      155.0000    0.9892
      160.0000    0.9718
      165.0000    0.9362
      170.0000    0.8736
      175.0000    0.7785
      180.0000    0.6532  
      185.0000    0.5098
      190.0000    0.3663
      195.0000    0.2405
      200.0000    0.1435
      205.0000    0.0776
      210.0000    0.0379
      215.0000    0.0167
      220.0000    0.0067
      225.0000    0.0024
      230.0000    0.0008
      235.0000    0.0002
      240.0000    0.0001
      245.0000    0.0000
      250.0000    0.0000

    Exercise \(\PageIndex{16}\)

    A junction point in a network has two incoming lines and two outgoing lines. The number of incoming messages \(N_1\) on line one in one hour is Poisson (50); on line 2 the number is \(N_2\) ~ Poisson (45). On incoming line 1 the messages have probability \(P_{1a} = 0.33\) of leaving on outgoing line a and \(1 - p_{1a}\) of leaving on line b. The messages coming in on line 2 have probability \(p_{2a} = 0.47\) of leaving on line a. Under the usual independence assumptions, what is the distribution of outgoing messages on line a? What are the probabilities of at least 30, 35, 40 outgoing messages on line a?

    Answer
    m1a = 50*0.33;  m2a = 45*0.47; ma = m1a + m2a;
    PNa = cpoisson(ma,[30 35 40])
    PNa =   0.9119    0.6890    0.3722

    Exercise \(\PageIndex{17}\)

    A computer store sells Macintosh, HP, and various other IBM compatible personal computers. It has two major sources of customers:

    1. Students and faculty from a nearby university
    2. General customers for home and business computing. Suppose the following assumptions are reasonable for monthly purchases.
    • The number of university buyers \(N1\) ~ Poisson (30). The probabilities for Mac, HP, others are 0.4, 0.2, 0.4, respectively.
    • The number of non-university buyers \(N2\) ~ Poisson (65). The respective probabilities for Mac, HP, others are 0.2, 0.3, 0.5.
    • For each group, the composite demand assumptions are reasonable, and the two groups buy independently.

    What is the distribution for the number of Mac sales? What is the distribution for the total number of Mac and Dell sales?

    Answer

    Mac sales Poisson (30*0.4 + 65*0.2 = 25); HP sales Poisson (30*0.2 + 65*0.3 = 25.5); total Mac plus HP sales Poisson(50.5).

    Exercise \(\PageIndex{18}\)

    The number \(N\) of “hits” in a day on a Web site on the internet is Poisson (80). Suppose the probability is 0.10 that any hit results in a sale, is 0.30 that the result is a request for information, and is 0.60 that the inquirer just browses but does not identify an interest. What is the probability of 10 or more sales? What is the probability that the number of sales is at least half the number of information requests (use suitable simple approximations)?

    Answer
    X = 0:30;
    Y = 0:80;
    PX = ipoisson(80*0.1,X);
    PY = ipoisson(80*0.3,Y);
    icalc:  X  Y  PX  PY
    - - - - - - - - - - - -
    PX10 = (X>=10)*PX'    % Approximate calculation
    PX10 =  0.2834
    pX10 = cpoisson(8,10)   % Direct calculation
    pX10 =  0.2834
    M = t>=0.5*u;
    PM = total(M.*P)
    PM =    0.1572

    Exercise \(\PageIndex{19}\)

    The number \(N\) of orders sent to the shipping department of a mail order house is Poisson (700). Orders require one of seven kinds of boxes, which with packing costs have distribution

    Cost (dollars) 0.75 1.25 2.00 2.50 3.00 3.50 4.00
    Probability 0.10 0.15 0.15 0.25 0.20 0.10 0.05

    What is the probability the total cost of the $2.50 boxes is no greater than $475? What is the probability the cost of the $2.50 boxes is greater than the cost of the $3.00 boxes? What is the probability the cost of the $2.50 boxes is not more than $50.00 greater than the cost of the $3.00 boxes? Suggestion. Truncate the Poisson distributions at about twice the mean value.

    Answer
    X = 0:400;
    Y = 0:300;
    PX = ipoisson(700*0.25,X);
    PY = ipoisson(700*0.20,Y);
    icalc
    Enter row matrix of X-values  X
    Enter row matrix of Y-values  Y
    Enter X probabilities  PX
    Enter Y probabilities  PY
     Use array operations on matrices X, Y, PX, PY, t, u, and P
    P1 = (2.5*X<=475)*PX'
    P1 =   0.8785
    M = 2.5*t<=(3*u + 50);
    PM = total(M.*P)
    PM =   0.7500

    Exercise \(\PageIndex{20}\)

    One car in 5 in a certain community is a Volvo. If the number of cars passing a traffic check point in an hour is Poisson (130), what is the expected number of Volvos? What is the probability of at least 30 Volvos? What is the probability the number of Volvos is between 16 and 40 (inclusive)?

    Answer
    P1 = cpoisson(130*0.2,30) = 0.2407
    P2 = cpoisson(26,16) - cpoisson(26,41) = 0.9819

    Exercise \(\PageIndex{21}\)

    A service center on an interstate highway experiences customers in a one-hour period as follows:

    • Northbound: Total vehicles: Poisson (200). Twenty percent are trucks.
    • Southbound: Total vehicles: Poisson (180). Twenty five percent are trucks.
    • Each truck has one or two persons, with respective probabilities 0.7 and 0.3.
    • Each car has 1, 2, 3, 4, or 5 persons, with probabilities 0.3, 0.3, 0.2, 0.1, 0.1, respectively

    Under the usual independence assumptions, let \(D\) be the number of persons to be served. Determine \(E[D]\), \(\text{Var} [D]\), and the generating function \(g_D (s)\).

    Answer

    \(T\) ~ Poisson (200*0.2 = 180*0.25 = 85), \(P\) ~ Poisson (200*0.8 + 180*0.75 = 295).

    a =   85
    b = 200*0.8 + 180*0.75
    b =  295
    YT = [1 2];
    PYT = [0.7 0.3];
    EYT = dot(YT,PYT)
    EYT =  1.3000
    VYT = dot(YT.^2,PYT) - EYT^2
    VYT =  0.2100
    YP = 1:5;
    PYP = 0.1*[3 3 2 1 1];
    EYP = dot(YP,PYP)
    EYP =  2.4000
    VYP = dot(YP.^2,PYP) - EYP^2
    VYP =   1.6400
    EDT = 85*EYT
    EDT =  110.5000  
    EDP = 295*EYP
    EDP =  708.0000
    ED = EDT + EDP
    ED =  818.5000
    VT = 85*(VYT + EYT^2)
    VT =  161.5000
    VP = 295*(VYP + EYP^2)
    VP =    2183
    VD = VT + VP
    VD =   2.2705e+03
     
    NT = 0:180;                   % Possible alternative
    gNT = ipoisson(85,NT);
    gYT = 0.1*[0 7 3];
    [DT,PDT] = gendf(gNT,gYT);
    EDT = dot(DT,PDT)
    EDT =  110.5000
    VDT = dot(DT.^2,PDT) - EDT^2
    VDT =  161.5000
    NP = 0:500;
    gNP = ipoisson(295,NP);
    gYP = 0.1*[0 3 2 2 1 1];
    [DP,PDP] = gendf(gNP,gYP);     %  Requires too much memory

    \(g_{DT} (s) = \text{exp} (85(0.7s + 0.3s^2 - 1))\) \(g_{DP} (s) = \text{exp} (295(0.1(3s + 3s^2 2s^3 + s^4 + s^5) - 1))\)

    \(g_D (s) = g_{DT} (s) g_{DP} (s)\)

    Exercise \(\PageIndex{22}\)

    The number \(N\) of customers in a shop in a given day is Poisson (120). Customers pay with cash or by MasterCard or Visa charge cards, with respective probabilties 0.25, 0.40, 0.35. Make the usual independence assumptions. Let \(N_1, N_2, N_3\) be the numbers of cash sales, MasterCard charges, Visa card charges, respectively. Determine \(P(N_1 \ge 30)\), \(P(N_2 \ge 60)\), \(P(N_3 \ge 50\), and \(P(N_2 > N_3)\).

    Answer
    X = 0:120;
    PX = ipoisson(120*0.4,X);
    Y = 0:120;
    PY = ipoisson(120*0.35,Y);
    icalc
    Enter row matrix of X values  X
    Enter row matrix of Y values  Y
    Enter X probabilities  PX
    Enter Y probabilities  PY
    Use array opertions on matrices X, Y, PX, PY, t, u, and P
    M = t > u;
    PM = total(M.*P)
    PM =    0.7190

    Exercise \(\PageIndex{23}\)

    A discount retail store has two outlets in Houston, with a common warehouse. Customer requests are phoned to the warehouse for pickup. Two items, a and b, are featured in a special sale. The number of orders in a day from store A is \(N_A\) ~ Poisson (30); from store B, the nember of orders is \(N_B\) ~ Poisson (40).

    For store A, the probability an order for a is 0.3, and for b is 0.7.

    For store B, the probability an order for a is 0.4, and for b is 0.6. What is the probability the total order for item b in a day is 50 or more?

    Answer

    P = cpoisson(30*0.7+40*0.6,50) = 0.2468

    Exercise \(\PageIndex{24}\)

    The number of bids on a job is a random variable \(N\) ~ binomial (7, 0.6). Bids (in thousands of dollars) are iid with \(Y\) uniform on [3, 5]. What is the probability of at least one bid of $3,500 or less? Note that “no bid” is not a bid of 0.

    Answer
    % First solution ---  FY(t) = 1 - gN[P(Y>t)]
    P = 1-(0.4 + 0.6*0.75)^7
    P  =    0.6794
    % Second solution --- Positive number of satisfactory bids,
    % i.e. the outcome is indicator for event E, with P(E) = 0.25
    pN = ibinom(7,0.6,0:7);
    gY = [3/4 1/4];         % Generator function for indicator
    [D,PD] = gendf(pN,gY);  % D is number of successes
    Pa = (D>0)*PD'          % D>0 means at least one successful bid
    Pa =    0.6794

    Exercise \(\PageIndex{25}\)

    The number of customers during the noon hour at a bank teller's station is a random number \(N\) with distribution

    \(N =\) 1 : 10, \(PN =\) 0.01 * [5 7 10 11 12 13 12 11 10 9]

    The amounts they want to withdraw can be represented by an iid class having the common distribution \(Y\) ~ exponential (0.01). Determine the probabilities that the maximum withdrawal is less than or equal to \(t\) for \(t = 100, 200, 300, 400, 500\).

    Answer

    Use \(F_W (t) = g_N[P(Y \le T)]\)

    gN = 0.01*[0 5 7 10 11 12 13 12 11 10 9];
    t = 100:100:500;
    PY = 1 - exp(-0.01*t);
    FW = polyval(fliplr(gN),PY)  % fliplr puts coeficients in
                                 % descending order of powers
    FW =    0.1330    0.4598    0.7490    0.8989    0.9615

    Exercise \(\PageIndex{26}\)

    A job is put out for bids. Experience indicates the number \(N\) of bids is a random variable having values 0 through 8, with respective probabilities

    Value 0 1 2 3 4 5 6 7 8
    Probability 0.05 0.10 0.15 0.20 0.20 0.10 0.10 0.07 0.03

    The market is such that bids (in thousands of dollars) are iid, uniform [100, 200]. Determine the probability of at least one bid of $125,000 or less.

    Answer

    Probability of a successful bid \(PY = (125 - 100)/100 = 0.25\)

    PY =0.25;
    gN = 0.01*[5 10 15 20 20 10 10 7 3];
    P = 1 - polyval(fliplr(gN),PY)
    P =  0.9116

    Exercise \(\PageIndex{27}\)

    A property is offered for sale. Experience indicates the number \(N\) of bids is a random variable having values 0 through 10, with respective probabilities

    Value 0 1 2 3 4 5 6 7 8 9 10
    Probability 0.05 0.15 0.15 0.20 0.10 0.10 0.05 0.05 0.05 0.05 0.05

    The market is such that bids (in thousands of dollars) are iid, uniform [150, 200] Determine the probability of at least one bid of $180,000 or more.

    Answer

    Consider a sequence of \(N\) trials with probabilty \(p = (180 - 150)/50 = 0.6\).

    gN = 0.01*[5 15 15 20 10 10 5 5 5 5 5];
    gY = [0.4 0.6];
    [D,PD] = gendf(gN,gY);
    P = (D>0)*PD'
    P =   0.8493

    Exercise \(\PageIndex{28}\)

    A property is offered for sale. Experience indicates the number \(N\) of bids is a random variable having values 0 through 8, with respective probabilities

    Number 0 1 2 3 4 5 6 7 8
    Probability 0.05 0.15 0.15 0.20 0.15 0.10 0.10 0.05 0.05

    The market is such that bids (in thousands of dollars) are iid symmetric triangular on [150 250]. Determine the probability of at least one bid of $210,000 or more.

    Answer
    gN = 0.01*[5 15 15 20 15 10 10 5 5];
    PY = 0.5 + 0.5*(1 - (4/5)^2)
    PY = 0.6800
    >> PW = 1 - polyval(fliplr(gN),PY)
    PW = 0.6536
    %alternate
    gY = [0.68 0.32];
    [D,PD] = gendf(gN,gY);
    P = (D>0)*PD'
    P = 0.6536

    Exercise \(\PageIndex{29}\)

    Suppose \(N\) ~ binomial (10, 0.3) and the \(Y_i\) are iid, uniform on [10, 20]. Let \(V\) be the minimum of the \(N\) values of the \(Y_i\). Determine \(P(V > t)\) for integer values from 10 to 20.

    Answer
    gN = ibinom(10,0.3,0:10);
    t = 10:20;
    p = 0.1*(20 - t);
    P = polyval(fliplr(gN),p) - 0.7^10
    P =
      Columns 1 through 7
        0.9718    0.7092    0.5104    0.3612    0.2503    0.1686    0.1092
      Columns 8 through 11
        0.0664    0.0360    0.0147         0
    Pa = (0.7 + 0.3*p).^10 - 0.7^10     % Alternate form of gN
    Pa =
      Columns 1 through 7
        0.9718    0.7092    0.5104    0.3612    0.2503    0.1686    0.1092
      Columns 8 through 11
        0.0664    0.0360    0.0147         0

    Exercise \(\PageIndex{30}\)

    Suppose a teacher is equally likely to have 0, 1, 2, 3 or 4 students come in during office hours on a given day. If the lengths of the individual visits, in minutes, are iid exponential (0.1), what is the probability that no visit will last more than 20 minutes.

    Answer
    gN = 0.2*ones(1,5);
    p = 1 - exp(-2);
    FW = polyval(fliplr(gN),p)
    FW =    0.7635
    gY = [p 1-p];               % Alternate
    [D,PD] = gendf(gN,gY);
    PW = (D==0)*PD'
    PW =    0.7635

    Exercise \(\PageIndex{31}\)

    Twelve solid-state modules are installed in a control system. If the modules are not defective, they have practically unlimited life. However, with probability \(p = 0.05\) any unit could have a defect which results in a lifetime (in hours) exponential (0.0025). Under the usual independence assumptions, what is the probability the unit does not fail because of a defective module in the first 500 hours after installation?

    Answer
    p = 1 - exp(-0.0025*500);
    FW = (0.95 + 0.05*p)^12
    FW =   0.8410
    gN = ibinom(12,0.05,0:12);
    gY = [p 1-p];
    [D,PD] = gendf(gN,gY);
    PW = (D==0)*PD'
    PW =   0.8410

    Exercise \(\PageIndex{32}\)

    The number \(N\) of bids on a painting is binomial (10, 0.3). The bid amounts (in thousands of dollars) \(Y_i\) form an iid class, with common density function \(f_Y (t) =0.005 (37 - 2t), 2 \le t \le 10\). What is the probability that the maximum amount bid is greater than $5,000?

    Answer

    \(P(Y \le 5) = 0.005 \int_{2}^{5} (37 - 2t)\ dt = 0.45\)

    p = 0.45;
    P = 1 - (0.7 + 0.3*p)^10
    P =   0.8352
    gN = ibinom(10,0.3,0:10);
    gY = [p 1-p];
    [D,PD] = gendf(gN,gY);  % D is number of "successes"
    Pa = (D>0)*PD'
    Pa =  0.8352

    Exercise \(\PageIndex{33}\)

    A computer store offers each customer who makes a purchase of $500 or more a free chance at a drawing for a prize. The probability of winning on a draw is 0.05. Suppose the times, in hours, between sales qualifying for a drawing is exponential (4). Under the usual independence assumptions, what is the expected time between a winning draw? What is the probability of three or more winners in a ten hour day? Of five or more?

    Answer

    \(N_t\) ~ Poisson (\(\lambda t\)), \(N_{Dt}\) ~ Poisson (\(\lambda pt\)), \(W_{Dt}\) exponential (\(\lambda p\)).

    p = 0.05;
    t = 10;
    lambda = 4;
    EW = 1/(lambda*p)
    EW =    5
    PND10 = cpoisson(lambda*p*t,[3 5])
    PND10 =  0.3233    0.0527

    Exercise \(\PageIndex{34}\)

    Noise pulses arrrive on a data phone line according to an arrival process such that for each \(t > 0\) the number \(N_t\) of arrivals in time interval \((0, t]\), in hours, is Poisson \((7t)\). The \(i\)th pulse has an “intensity” \(Y_i\) such that the class \(\{Y_i: 1 \le i\}\) is iid, with the common distribution function \(F_Y (u) = 1 - e^{-2u^2}\) for \(u \ge 0\). Determine the probability that in an eight-hour day the intensity will not exceed two.

    Answer

    \(N_8\) is Poisson (7*8 = 56) \(g_N (s) = e^{56(s - 1)}\).

    t = 2;
    FW2 = exp(56*(1 - exp(-t^2) - 1))
    FW2 =   0.3586

    Exercise \(\PageIndex{35}\)

    The number \(N\) of noise bursts on a data transmission line in a period \((0, t]\) is Poisson (\(\mu\)). The number of digit errors caused by the \(i\)th burst is \(Y_i\), with the class \(\{Y_i: 1 \le i\}\) iid, \(Y_i - 1\) ~ geometric \((p)\). An error correcting system is capable or correcting five or fewer errors in any burst. Suppose \(\mu = 12\) and \(p = 0.35\). What is the probability of no uncorrected error in two hours of operation?

    Answer

    \(F_W (k) = g_N [P(Y \le k)]P(Y \le k) - 1 - q^{k - 1}\ \ N_t\) ~ Poisson (12\(t\))

    q = 1 - 0.35;
    k = 5;
    t = 2;
    mu = 12;
    FW = exp(mu*t*(1 - q^(k-1) - 1))
    FW =  0.0138
    
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