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Consider a Bernoulli sequence with probability p of success on any component trial. Let N be the number of the trial on which the first success occurs. Let Y _{i} be the time (or cost) to execute the i th trial. Then the total time (or cost) from the beginning to the completion of the first success is
We suppose the Y _{i} form an iid class, independent of N . Now $N-1\sim $ geometric $\left(p\right)$ implies
${g}_{N}\left(s\right)=ps/(1-qs)$ , so that
There are two useful special cases:
Suppose a prospective employer is interviewing candidates for a job from a pool in which twenty percent are qualified. Interview times (in hours) Y _{i} are presumed to form an iid class, each exponential (3). Thus, the average interview time is 1/3 hour (twentyminutes). We take the probability for success on any interview to be $p=0.2$ . What is the probability a satisfactory candidate will be found in four hours or less?What is the probability the maximum interview time will be no greater than 0.5, 0.75, 1, 1.25, 1.5 hours?
SOLUTION
$T\sim $ exponential $(0.2\xb73=0.6)$ , so that $P(T\le 4)=1-{e}^{-0.6\xb74}=0.9093$ .
MATLAB computations give
t = 0.5:0.25:1.5;
PWt = (1 - exp(-3*t))./(1 + 4*exp(-3*t));disp([t;PWt]')0.5000 0.4105
0.7500 0.62931.0000 0.7924
1.2500 0.89251.5000 0.9468
The average interview time is 1/3 hour; with probability 0.63 the maximum is 3/4 hour or less; with probability 0.79 the maximum is one hour or less; etc.
In the general case, solving for the distribution of T requires transform theory, and may be handled best by a program such as Maple or Mathematica.
For the case of simple Y _{i} , we may use approximation procedures based on properties of the geometric series. Since $N-1\sim $ geometric $\left(p\right)$ ,
Note that ${g}_{n}\left(s\right)$ has the form of the generating function for a simple approximation N _{n} which matches values and probabilities with N up to $k=n$ . Now
The evaluation involves convolution of coefficients which effectively sets $s=1$ . Since ${g}_{N}\left(1\right)={g}_{Y}\left(1\right)=1$ ,
which is negligible if n is large enough. Suitable n may be determined in each case. With such an n , if the Y _{i} are nonnegative, integer-valued, we may use the gend procedure on ${g}_{n}\left[{g}_{Y}\left(s\right)\right]$ , where
For the integer-valued case, as in the general case of simple Y _{i} , we could use mgd. However, gend is usually faster and more efficient for the integer-valued case. Unless q is small, the number of terms needed to approximate g _{n} is likely to be too great.
Let $p=0.3$ and Y be uniformly distributed on $\{1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}10\}$ . Determine the distribution for
SOLUTION
p = 0.3;
q = 1 - p;a = [30 35 40]; % Check for suitable nb = q.^a
b = 1.0e-04 * % Use n = 400.2254 0.0379 0.0064
n = 40;k = 1:n;
gY = 0.1*[0 ones(1,10)];
gN = p*[0 q.^(k-1)]; % Probabilities, 0<= k<= 40
gendDo not forget zero coefficients for missing powers
Enter gen fn COEFFICIENTS for gN gNEnter gen fn COEFFICIENTS for gY gY
Values are in row matrix D; probabilities are in PD.To view the distribution, call for gD.
sum(PD) % Check sum of probabilitiesans = 1.0000
FD = cumsum(PD); % Distribution function for Dplot(0:100,FD(1:101)) % See
[link] P50 = (D<=50)*PD'
P50 = 0.9497P30 = (D<=30)*PD'
P30 = 0.8263
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