CE93-Engineering Data Analysis
Joan Walker, Spring 2011
Assignment 6
Due 3/9/2011 11:10 AM
1.
Consider the following game: A fair coin is flipped until the first tail appears; we win $2 if it appears
on the first toss, $4 if it appears on the second toss, and, in general, $2
k
if it first occurs on the
k
th
toss. Let the random variable X denote our winnings. How much should we have to pay to play in
order for this to be a fair game?
(Note
:
A fair game is one where the difference between the payment required to play the game and
E(X) equals 0). You may find it helpful to first find the probability mass function of the number of
tosses needed to reach
The easiest way to conceptualize this problem is to break the problem down into a smaller
problem
–
if you limited the # flips you can have to 1 flip, then you’d get an E(X) = 1 (assuming
that a heads gives you $0 in winnings). Then extend it to the 2-flip case (HH, HT, and T), then 3-
flip (HHH, HHT, HT, and T), and then generalize it into an infinite number of flips’ case. Each
additional flip allowed gives you +1 more to E(X), so if an infinite number of flips is allowed (i.e.
un
til you finally get tails), you’d get E(X) = infinity
The Math:
The probability that the first tail appears on the first toss is ½, on the second toss ¼, and, in general,
1
2
°
, where k is the number of tosses. This can be expressed as:
?
±
°²
=
1
2
°
Thus:
³ ?²
=
´µ
$2
¶
·
∞
¶
=1
?
±
¶²
=
´µ
$2
¶
·
∞
¶
=1
1
2
¶
= 1 + 1 + 1 +
⋯
=
∞
This is also known as the “St. Petersburg” paradox.
2.
A wastewater treatment plant (WWTP) treats water from primarily two distinct sources, one from
stormwater, and two from human sources (including residential, commercial, and industrial
wastewater). Let X be the amount of flow of wastewater from stormwater, and let Y be the amount