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The class of random variables is iid (independent, identically distributed) with common distribution
Let . Determine the distribution for W and from this determine and . Do this with icalc, then repeat with icalc3 and compare results.
Since icalc uses X and in its output, we avoid a renaming problem by using x and for data vectors X and .
x = [-5 -1 3 4 7];px = 0.01*[15 20 30 25 10];icalc
Enter row matrix of X-values 3*xEnter row matrix of Y-values -4*x
Enter X probabilities pxEnter Y probabilities px
Use array operations on matrices X, Y, PX, PY, t, u, and Pa = t + u;
[V,PV]= csort(a,P);
icalcEnter row matrix of X-values V
Enter row matrix of Y-values 2*xEnter X probabilities PV
Enter Y probabilities pxUse array operations on matrices X, Y, PX, PY, t, u, and P
b = t + u;[W,PW] = csort(b,P);P1 = (W>0)*PW'
P1 = 0.5300P2 = ((-20<=W)&(W<=10))*PW'
P2 = 0.5514icalc3 % Alternate using icalc3
Enter row matrix of X-values xEnter row matrix of Y-values x
Enter row matrix of Z-values xEnter X probabilities px
Enter Y probabilities pxEnter Z probabilities px
Use array operations on matrices X, Y, Z,PX, PY, PZ, t, u, v, and P
a = 3*t - 4*u + 2*v;[W,PW] = csort(a,P);P1 = (W>0)*PW'
P1 = 0.5300P2 = ((-20<=W)&(W<=10))*PW'
P2 = 0.5514 The class is independent; the respective probabilites for these events are . Consider the simple random variables
Determine , , .
cx = [3 -9 4 0];pmx = minprob(0.01*[42 27 33]);cy = [-2 6 2 -3];pmy = minprob(0.01*[47 37 41]);[X,PX] = canonicf(cx,pmx);[Y,PY] = canonicf(cy,pmy);icalc
Enter row matrix of X-values XEnter row matrix of Y-values Y
Enter X probabilities PXEnter Y probabilities PY
Use array operations on matrices X, Y, PX, PY, t, u, and PG = 2*t - 3*u;
[Z,PZ]= csort(G,P);
PYgX = total((u>t).*P)
PYgX = 0.3752PZpos = (Z>0)*PZ'
PZpos = 0.5654P5Z25 = ((5<=Z)&(Z<=25))*PZ'
P5Z25 = 0.4745 Two players, Ronald and Mike, throw a pair of dice 30 times each. What is the probability Mike throws more “sevens” than does Ronald?
P = (ibinom(30,1/6,0:29))*(cbinom(30,1/6,1:30))' = 0.4307
A class has fifteen boys and fifteen girls. They pair up and each tosses a coin 20 times. What is the probability that at least eight girls throw more heads thantheir partners?
pg = (ibinom(20,1/2,0:19))*(cbinom(20,1/2,1:20))'
pg = 0.4373 % Probability each girl throws moreP = cbinom(15,pg,8)
P = 0.3100 % Probability eight or more girls throw more Glenn makes five sales calls, with probabilities , , , , , of success on the respective calls. Margaret makes four sales calls with probabilities , , , , of success on the respective calls. Assume that all nine events form an independent class. If Glenn realizes a profit of $18.00 on each sale and Margaret earns$20.00 on each sale, what is the probability Margaret's gain is at least $10.00 more thanGlenn's?
cg = [18*ones(1,5) 0];cm = [20*ones(1,4) 0];pmg = minprob(0.01*[37 52 48 71 63]);pmm = minprob(0.01*[77 82 75 91]);[G,PG] = canonicf(cg,pmg);[M,PM] = canonicf(cm,pmm);icalc
Enter row matrix of X-values GEnter row matrix of Y-values M
Enter X probabilities PGEnter Y probabilities PM
Use array operations on matrices X, Y, PX, PY, t, u, and PH = u-t>=10;
p1 = total(H.*P)p1 = 0.5197 Mike and Harry have a basketball shooting contest.
Let be the number of points scored by Mike and Harry, respectively. Determine , and .
X = 2*[0:10];PX = ibinom(10,0.75,0:10);
Y = 3*[0:12];
PY = ibinom(12,0.40,0:12);icalc
Enter row matrix of X-values XEnter row matrix of Y-values Y
Enter X probabilities PXEnter Y probabilities PY
Use array operations on matrices X, Y, PX, PY, t, u, and PPX15 = (X>=15)*PX'
PX15 = 0.5256PY15 = (Y>=15)*PY'
PY15 = 0.5618G = t>=u;
PG = total(G.*P)PG = 0.5811 Martha has the choice of two games.
Martha has $100 to bet. She is trying to decide whether to play Game 1 ten times or Game 2 twenty times. Let and be the respective net winnings (payoff minus fee to play).
Which game seems preferable?
W1 = 20*[0:10] - 100;PW1 = ibinom(10,1/2,0:10);
W2 = 15*[0:20]- 100;
PW2 = ibinom(20,1/3,0:20);P1pos = (W1>0)*PW1'
P1pos = 0.3770P2pos = (W2>0)*PW2'
P2pos = 0.5207icalc
Enter row matrix of X-values W1Enter row matrix of Y-values W2
Enter X probabilities PW1Enter Y probabilities PW2
Use array operations on matrices X, Y, PX, PY, t, u, and PG = u>= t;
PG = total(G.*P)PG = 0.5182 Jim and Bill of the men's basketball team challenge women players Mary and Ellen to a free throw contest. Each takes five free throws.Make the usual independence assumptions. Jim, Bill, Mary, and Ellen have respective probabilities , and 0.85 of making each shot tried. What is the probability Mary and Ellen make a total number of free throwsat least as great as the total made by the guys?
x = 0:5;
PJ = ibinom(5,0.82,x);PB = ibinom(5,0.87,x);
PM = ibinom(5,0.80,x);PE = ibinom(5,0.85,x);icalc
Enter row matrix of X-values xEnter row matrix of Y-values x
Enter X probabilities PJEnter Y probabilities PB
Use array operations on matrices X, Y, PX, PY, t, u, and PH = t+u;
[Tm,Pm]= csort(H,P);
icalcEnter row matrix of X-values x
Enter row matrix of Y-values xEnter X probabilities PM
Enter Y probabilities PEUse array operations on matrices X, Y, PX, PY, t, u, and P
G = t+u;[Tw,Pw] = csort(G,P);icalc
Enter row matrix of X-values TmEnter row matrix of Y-values Tw
Enter X probabilities PmEnter Y probabilities Pw
Use array operations on matrices X, Y, PX, PY, t, u, and PGw = u>=t;
PGw = total(Gw.*P)PGw = 0.5746icalc4 % Alternate using icalc4
Enter row matrix of X-values xEnter row matrix of Y-values x
Enter row matrix of Z-values xEnter row matrix of W-values x
Enter X probabilities PJEnter Y probabilities PB
Enter Z probabilities PMEnter W probabilities PE
Use array operations on matrices X, Y, Z,WPX, PY, PZ, PW t, u, v, w, and P
H = v+w>= t+u;
PH = total(H.*P)PH = 0.5746 
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