Stat 231 Exam 1
Fall 2010
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4. A password generator generates passwords of length 7 characters. These characters may be
lower case letters (a through z), upper case letters (A through Z), or digits (0 through 9).
5 pts
a) How many different passwords can be generated?
5 pts
b) How many passwords begin with a lower case letter, end with an upper case letter, and contain
at least 1 digit?
Suppose henceforth that passwords are randomly generated in the sense that each possible password
is equally likely to be generated and successive passwords are independent.
5 pts
c) My password is juL765r . A hacker is randomly generating passwords trying to guess mine.
What is the expected number of attempts the hacker will have to make in order to get my password?
5 pts
d) What is the conditional probability that the next password generated contains at least 1 digit
given that it begins with a lower case letter and ends with an upper case letter?
5
10 pts
10 pts
5. Engineering requirements are that a diameter of a cylinder are 2.500 inch .001 inch . A lathe
cuts cylinders whose diameters are normally distributed with mean
2.500 inch and standard
deviation inch. How small does need to be so that 95% of cylinders cut on the lathe meet
engineering requirements?
6. A so-called " k out of n " system will function provided at least k of its n components function.
Consider a "4 out of 5" system with independent components that each have reliability (probability
of functioning) p . I need to know how large p needs to be in order to have system reliability
(probability of functioning) .99. Set up an equation you would need to solve in order to answer this
for me.
6
8 pts
2. A fair tetrahedronal die (with 4 faces numbered 1,2,3, and 4 equally likely to be turned down
upon throwing) is tossed and a fair coin is tossed. If the coin lands tails up, 1 is added to the value
read from the die and if it lands heads up, 1 is subtracted from the value read from the die. Let
X number on the "down" face of the tetrahedronal die
and
Y the final sum of X and the appropriate one of 1 or 1
a) Below is a joint probability table for the pair X , Y . Fill in the values of the joint pmf f x, y ,
and the marginal pmf's g x and h y .
y\x
1
2
3
4
h y
5
4
3
2
1
0
g x
Henceforth, use whatever you've written in answer to a) to answer b) and c).
6 pts
b) Are the two random variables X and Y independent? Very carefully explain why or why not.
("yes" or "no" without explanation is not a sufficient answer.)
6 pts
c) Find the mean value for Y , EY , and also the mean of the conditional distribution of Y given
that X 2 .
EY
_____________
mean of the conditional distribution of Y given X
2 ___________
3
1. Uniform (pseudo-) random number generators supposedly produce realizations of independent
random variables with marginal probability densities
1 if 0 u 1
f u
0 otherwise
Suppose that X and Y are the next two "random numbers" generated by my calculator. The joint
pdf for these variables is then
1 if 0 x 1 and 0 y 1
f x, y
0 otherwise
7 pts
a) Find both P X
Y
.5 and P X
Y
1| X
.5 . (You may want to draw a figure reminding
yourself where the joint pdf is positive. Neither of these values should require much calculation.)
P X
7 pts
Y
.5
____________
b) For first 0 t 1 and then 1 t
For 0 t 1 : ____________
6 pts
P X
Y
1| X
.5
____________
2 , evaluate (as a function of t ) P X
For 1 t
c) Find the probability density for the random variable T
X
Y
t .
2 : ____________
Y , f t . (Be sure to say very
carefully where any expression(s) you write involving t are relevant.)
2
3. The actual dimensions W and H (width and height) of some rectangular cut sheets of poster
board are random. Suppose that the mean width is 48.00 inches and mean height is 36.00 inches ,
and that the standard deviations of both dimensions are .05 inch . Suppose further that the cutting
of these sheets is such that W and H may be modeled as independent random variables.
10 pts
a) Find (exactly) the standard deviation of the area, A WH , of a sheet. (Hint: laws of expectation
and variance)
10 pts
b) Find a ("propagation of error") approximation for the standard deviation of the aspect ratio,
R W / H , of a sheet.
4