15.1 Some random selection problems  (Page 3/6)

Lowest bidder

A manufacturer seeks bids on a modification of one of his processing units. Twenty contractors are invited to bid. They bid with probability 0.3, so that the numberof bids $N\sim$ binomial (20,0.3). Assume the bids Y _i (in thousands of dollars) form an iid class. The market is such that the bids have a common distributionsymmetric triangular on (150,250). What is the probability of at least one bid no greater than 170, 180, 190, 200, 210? Note that no bid is not a low bid of zero, hence we must use the special case.

Solution

$$P(V\le t)=1-g_N\left[P(Y>t)\right]=1-{(0.7+0.3p)}^{20}\text{where}p=P(Y>t)$$

Solving graphically for $p=P(V>t)$ , we get

$$p=[23/2541/5017/251/28/25]\text{for}t=\left[170180190200210\right]$$

Now $g_N\left(s\right)={(0.7+0.3s)}^{20}$ . We use MATLAB to obtain

[link] With a general counting variable

Suppose the number of bids is 1, 2 or 3 with probabilities 0.3, 0.5, 0.2, respectively.

Determine $P(V\le t)$ in each case.

SOLUTION.

The minimum of the selected $Y^{\text{'}}s$ is no greater than t if and only if there is at least one Y less than or equal to t . We determine in each case probabilities for the number of bids satisfying $Y\le t$ . For each t , we are interested in the probability of one or more occurrences of the event $Y\le t$ . This is essentially the problem in Example 7 from "Random Selection", with probability $p=P(Y\le t)$ .

[link] may be worked in this manner by using gN = ibinom(20,0.3,0:20) . The results, of course, are the same as in the previous solution. The fact that the probabilities in this example are lower for each t than in [link] reflects the fact that there are probably fewer bids in each case.

Batch testing

Electrical units from a production line are first inspected for operability. However, experience indicates that a fraction p of those passing the initial operability test are defective. All operable units are subsequenly tested in a batch under continuousoperation ( a “burn in” test). Statistical data indicate the defective units have times to failure Y _i iid, exponential $\left(\lambda \right)$ , whereas good units have very long life (infinite from the point of view of the test). A batch of n units is tested. Let V be the time of the first failure and N be the number of defective units in the batch. If the test goes t units of time with no failure (i.e., $V>t$ ), what is the probability of no defective units?

SOLUTION

Since no defective units implies no failures in any reasonable test time, we have

Since $N=0$ does not yield a minimum value, we have $P(V>t)=g_N\left[P(Y>t)\right]$ . Now under the condition above, the number of defective units $N\sim$ binomial $(n,p)$ , so that $g_N\left(s\right)={(q+ps)}^n$ . If N is large and p is reasonably small, N is approximately Poisson $\left(np\right)$ with $g_N\left(s\right)=e^{np(s-1)}$ and $P(N=0)=e^{-np}$ . Now $P(Y>t)=e^{-\lambda t}$ ; for large n

For $n=5000,p=0.001,\lambda =2$ , and $t=1,2,3,4,5$ , MATLAB calculations give

It appears that a test of three to five hours should give reliable results. In actually designing the test, one should probably make calculations with a number of differentassumptions on the fraction of defective units and the life duration of defective units. These calculations are relatively easy to make with MATLAB.