Spring 2015
Marmara University
Electrical and Electronics Engineering
STAT253-Probability and Statistics
Computer Homework 1
A manufacturer of electronic memory chips produces batches of 1000 chips for shipment to computer
companies. To determine if the chips meet specifications the manufacturer initially tests all 1000 chips in each
batch. As demand for the chips grows, however, he realizes that it is impossible to test all the chips and so
proposes that only a subset or sample of the batch be tested. The criterion for acceptance of the batch is that at
least N % of the sample chips tested meet specifications. If the criterion is met, then the batch is accepted and
shipped. This criterion is based on past experience of what the computer companies will find acceptable, i.e., if
the batch "yield" is less than N % the computer companies will not be happy. The production manager proposes
that a sample of 100 chips from the batch be tested and if N or more are deemed to meet specifications, then the
batch is judged to be acceptable. However, a quality control supervisor argues that even if only 100 - N of the
sample chips are defective, then it is still quite probable that the batch will not have a N % yield and thus be
defective.
The quality control supervisor wishes to convince the production manager that a defective batch can
frequently produce 100 - N or fewer defective chips in a chip sample of size 100. He does so by determining the
probability that a defective batch will have a chip sample with 100 - N or fewer defective chips as follows. He
first needs to assume the proportion of chips in the defective batch that will be good. Since a good batch has a
proportion of good chips of N %, a defective batch will have a proportion of good chips of less than N %. Then,
according to the production manager a batch is judged to be acceptable if the sample produces N, N + 1, N + 2,
…, 99, or 100 good chips. The quality control supervisor likens this problem to the drawing of 100 balls from an
“chip urn” containing 1000 balls. In the urn there are 1000p good balls and 1000(1 — p) bad ones. The
probability of drawing N or more good balls from the urn is given approximately by the binomial probability
law.
a)
Let N = 95. Now the defective batch will be judged as acceptable if there are 95 or more successes out of a
possible 100 draws. The probability of this occurring is
100
100 𝑘
𝑃[𝑘 ≥ 95] = ∑ (
) 𝑝 (1 − 𝑝)100−𝑘
𝑘
𝑘=95
Plot probability of accepting a defective batch or 𝑃[𝑘 ≥ 95] as function of p for p = 0.9, 0.91, 0.92, 0.93,
0.94, 0.95. Is the quality control supervisor indeed correct? For example, what is 𝑃[𝑘 ≥ 95] for p = 0.94?
Elaborate on it.
b) Let N = 98. Now the defective batch will be judged as acceptable if there are 98 or more successes out of a
possible 100 draws. The probability of this occurring is
100
100 𝑘
𝑃[𝑘 ≥ 98] = ∑ (
) 𝑝 (1 − 𝑝)100−𝑘
𝑘
𝑘=98
Plot 𝑃[𝑘 ≥ 98] as function of p for p = 0.9, 0.91, 0.92, 0.93, 0.94, 0.95. What is 𝑃[𝑘 ≥ 98] for p = 0.94?
Did the probability of accepting a defective batch increase or decrease compared to part a) ? Why is so?
Spring 2015
Marmara University
Electrical and Electronics Engineering
STAT253-Probability and Statistics
Computer Homework 2
An increasingly common health problem in many countries is obesity. It has been found to be associated with
many life-threatening illnesses, especially diabetes. One way to define what constitutes an obese person is via
the body mass index (BMI). It is computed as
BMI =
703 × 𝑊
𝐻2
where W is the weight of the person in pounds and H is the person's height in inches. BMIs greater than 25 and
less than 30 are considered to indicate an overweight person, and 30 and above an obese person. It is of great
importance to be able to estimate the probability mass function (PMF) of the BMI for a population of people. For
example, a table of the joint probabilities of heights and weights for a hypothetical population of college students
is given by
For this population we would like to know the probability or percentage of obese persons. This percentage of the
population would then be at risk for developing diabetes. To do so we could first determine the PMF of the BMI
and then determine the probability of a BMI of 30 and above. From the above table we have the joint PMF for
the random vector {H, W). To find the PMF for the BMI we note that it is a function of H and W, we wish to
determine the PMF of Z = g{X, Y), where Z denotes the BMI, X denotes the height, and Y denotes the weight.
The solution follows immediately from
𝑝𝑍 [𝑧𝑘 ] =
∑ ∑ 𝑃𝑋,𝑌 [𝑥𝑖 , 𝑦𝑗 ]
{(𝑖, 𝑗): 𝑧𝑘 = 𝑔(𝑥𝑖 , 𝑦𝑗 )}
One slight modification that we must make in order to fit the data of the above table into our theoretical
framework is to replace the height and weight intervals by their midpoint values. For example, in the table the
probability of observing a person with a height between 5'8" and 6' and a weight of between 130 and 160 lbs. is
0.06. We convert these intervals so that we can say the probability of a person having a height of 5'10"and a
weight of 145 lbs. is 0.06.
a) Determine the PMF of the BMI and show it as a stem plot.
b) What is the probability of being obese (BMI ≥ 30)?