ECE 313 Probability with Engineering Applications Fall 2007
Homework 5 Due Sept. 26, 2007
28. Consider an experiment of coin flips where each flip is a head with probability pand the flips
are independent. Let Xi= 1 if the ith flip is a head and Xi= 0 otherwise.
(a) Suppose p= 0.25. Plot the pmf of the random variable Y= 1/n Pn
i=1 Xifor n= 5,10,50,200.
Be sure you scale the axes appropriately so that it is meaningful to compare the different plots.
(b) You want to estimate Yby specifying an interval symmetric around E(Y) such that you can
be 99% sure that Ywill lie in that interval. Numerically calculate this interval for p= 0.25 and
n= 5,10,50,200. How does the size of this interval change with n? What do you think will happen
as n→ ∞?
(c) For n= 200, calculate the size of the interval in (b) for different values of p, and plot the interval
size against the variance of Y. Can you give an intuitive explanation for the trends in your graph?
29. (a) We are given a sample space Ω and a probability law P. Let Bbe an event with non-zero
probability. Define the real-valued function Qon the set of events in Ω by Q(A) := P(A|B). Show
that Qsatisfies the three axioms of a probability law and hence is a valid probability law.
(b) Using (a) or by direct calculations, show the the following formula:
P(A|B, C ) = P(C|A, B )P(A|B)
P(C|B)
30. A laboratory blood test is 95 percent effective in detecting a certain disease when it is in fact
present. However, the test also yields a false positive result for 1 percent of the healthy person
tested. (That is, if a healthy person is tested, then with probability 0.01, the test will imply that he
has the disease.) If 0.5 percent of the population actually has the disease, what is the probability
a person has the disease given that his test result is positive? Are you surprised by the answer?
Explain.
31. Lets consider the example in class where the document has nletters and the probability that
the kth letter is typed wrongly is p.
(a) Explain why there is not enough information to calculate the probability that the entire docu-
ment is typed correctly.
(b) Find the smallest possible value for this probability in terms of pand specify the probability
law for which this smallest value is achieved.
(c) Let n= 2. Suppose you are told additionally that q1is the conditional probability that the
second letter is typed wrongly given that the first letter is typed wrongly, and q2be the conditional
probability that the second letter is typed wrongly given that the first letter is typed correctly. Do
we now have a complete specification of the probability law?
(d) What constraint does q1and q2have to satisfy?
(e) Under what values of q1and q2is the probability that the document is typed correctly (i) equal
to, (ii) larger than, and (iii) smaller than the probability when the events that the two letters are
typed incorrectly are independent?
32. There are 2 machines having lifetimes distributed with pmf’s p1and p2. Suppose one of the 2
machines is randomly picked with equal probability and put in operation at time 0. Conditional
on the fact that the machine is still running at time , what is the probability that it is machine 1
that was picked?
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