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On a Tuesday evening, the Houston Rockets, the Orlando Magic, and the Chicago Bulls all have games (but not with one another). Let A be the event the Rockets win, B be the event the Magic win, and C be the event the Bulls win. Suppose the class is independent, with respective probabilities 0.75, 0.70 0.8. Ellen's boyfriend is a rabid Rockets fan, who does not likethe Magic. He wants to bet on the games. She decides to take him up on his bets as follows:
Ellen's winning may be expressed as the random variable
Determine the distribution for X . What are the probabilities Ellen loses money, breaks even, or comes out ahead?
P = 0.01*[75 70 80];c = [-15 15 -10 10];canonic
Enter row vector of coefficients cEnter row vector of minterm probabilities minprob(P)
Use row matrices X and PX for calculationsCall for XDBN to view the distribution
disp(XDBN)-15.0000 0.1800
-5.0000 0.04500 0.4800
10.0000 0.120015.0000 0.1400
25.0000 0.0350PXneg = (X<0)*PX'
PXneg = 0.2250PX0 = (X==0)*PX'
PX0 = 0.4800PXpos = (X>0)*PX'
PXpos = 0.2950 The class has minterm probabilities
npr06_12 Minterm probabilities in pm, coefficients in c
a = imintest(pm)The class is NOT independent
Minterms for which the product rule failsa =
1 1 1 11 1 1 1
1 1 1 11 1 1 1
canonicEnter row vector of coefficients c
Enter row vector of minterm probabilities pmUse row matrices X and PX for calculations
Call for XDBN to view the distributionXDBN =
0 0.00501.0000 0.0430
2.0000 0.21203.0000 0.4380
4.0000 0.3020P2 = (X>=2)*PX'
P2 = 0.9520P13 = ((X==1)|(X==3))*PX'
P13 = 0.4810 James is expecting three checks in the mail, for $20, $26, and $33 dollars. Their arrivals are the events . Assume the class is independent, with respective probabilities 0.90, 0.75, 0.80. Then
represents the total amount received. Determine the distribution for X . What is the probability he receives at least $50? Less than $30?
c = [20 26 33 0];P = 0.01*[90 75 80];canonic
Enter row vector of coefficients cEnter row vector of minterm probabilities minprob(P)
Use row matrices X and PX for calculationsCall for XDBN to view the distribution
disp(XDBN)0 0.0050
20.0000 0.045026.0000 0.0150
33.0000 0.020046.0000 0.1350
53.0000 0.180059.0000 0.0600
79.0000 0.5400P50 = (X>=50)*PX'
P50 = 0.7800P30 = (X<30)*PX'
P30 = 0.0650 A gambler places three bets. He puts down two dollars for each bet. He picks up three dollars (his original bet plus one dollar) if he wins the first bet, four dollarsif he wins the second bet, and six dollars if he wins the third. His net winning can be represented by the random variable
Assume the results of the games are independent. Determine the distribution for X .
c = [3 4 6 -6];P = 0.1*[5 4 3];canonic
Enter row vector of coefficients cEnter row vector of minterm probabilities minprob(P)
Use row matrices X and PX for calculationsCall for XDBN to view the distribution
dsp(XDBN)-6.0000 0.2100
-3.0000 0.2100-2.0000 0.1400
0 0.09001.0000 0.1400
3.0000 0.09004.0000 0.0600
7.0000 0.0600 Henry goes to a hardware store. He considers a power drill at $35, a socket wrench set at $56, a set of screwdrivers at $18, a vise at $24, and hammer at $8. Hedecides independently on the purchases of the individual items, with respective probabilities 0.5, 0.6, 0.7, 0.4, 0.9. Let X be the amount of his total purchases. Determine the distribution for X .
c = [35 56 18 24 8 0];P = 0.1*[5 6 7 4 9];canonic
Enter row vector of coefficients cEnter row vector of minterm probabilities minprob(P)
Use row matrices X and PX for calculationsCall for XDBN to view the distribution
disp(XDBN)0 0.0036
8.0000 0.032418.0000 0.0084
24.0000 0.002426.0000 0.0756
32.0000 0.021635.0000 0.0036
42.0000 0.005643.0000 0.0324
50.0000 0.050453.0000 0.0084
56.0000 0.005459.0000 0.0024
61.0000 0.075664.0000 0.0486
67.0000 0.021674.0000 0.0126
77.0000 0.005680.0000 0.0036
82.0000 0.113485.0000 0.0504
88.0000 0.032491.0000 0.0054
98.0000 0.008499.0000 0.0486
106.0000 0.0756109.0000 0.0126
115.0000 0.0036117.0000 0.1134
123.0000 0.0324133.0000 0.0084
141.0000 0.0756 A sequence of trials (not necessarily independent) is performed. Let E i be the event of success on the i th component trial. We associate with each trial a “payoff function” . Thus, an amount a is earned if there is a success on the trial and an amount b (usually negative) if there is a failure. Let S n be the number of successes in the n trials and W be the net payoff. Show that .
A marker is placed at a reference position on a line (taken to be the origin); a coin is tossed repeatedly. If a head turns up, the marker is moved one unit to the right;if a tail turns up, the marker is moved one unit to the left.
S = 0:10;
PS = ibinom(10,0.5,0:10);X = 2*S - 10;
disp([X;PS]')
-10.0000 0.0010-8.0000 0.0098
-6.0000 0.0439-4.0000 0.1172
-2.0000 0.20510 0.2461
2.0000 0.20514.0000 0.1172
6.0000 0.04398.0000 0.0098
10.0000 0.0010 Margaret considers five purchases in the amounts 5, 17, 21, 8, 15 dollars with respective probabilities 0.37, 0.22, 0.38, 0.81, 0.63. Anne contemplates sixpurchases in the amounts 8, 15, 12, 18, 15, 12 dollars, with respective probabilities 0.77, 0.52, 0.23, 0.41, 0.83, 0.58. Assume that all eleven possible purchases form anindependent class.
Suggestion for part (c). Let MATLAB perform the calculations.
[r,s] = ndgrid(X,Y);[t,u] = ndgrid(PX,PY);z = r + s;
pz = t.*u;[Z,PZ] = csort(z,pz); % file
npr06_18.m cx = [5 17 21 8 15 0];cy = [8 15 12 18 15 12 0];pmx = minprob(0.01*[37 22 38 81 63]);pmy = minprob(0.01*[77 52 23 41 83 58]);
npr06_18 [X,PX] = canonicf(cx,pmx); [Y,PY]= canonicf(cy,pmy);
[r,s]= ndgrid(X,Y); [t,u] = ndgrid(PX,PY);z = r + s; pz = t.*u;
[Z,PZ]= csort(z,pz);
a = length(Z)a = 125 % 125 different values
plot(Z,cumsum(PZ)) % See figure Plotting details omitted 
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