Discrete Mathematics and Probability Theory HW 11

CS 70 Discrete Mathematics and Probability Theory

HW 11
Sundry
Before you start your homework, write down your team. Who else did you work with on this
homework? List names and email addresses. (In case of homework party, you can also just describe
the group.) How did you work on this homework? Working in groups of 3-5 will earn credit for
your "Sundry" grade.
Please copy the following statement and sign next to it:
I certify that all solutions are entirely in my words and that I have not looked at another student’s
solutions. I have credited all external sources in this write up.
1 Proof with Indicators
Let n ∈ Z+. Let α1,...,αn ∈ R and let A1,...,An be events. Prove that ∑
n
i=1 ∑
n
j=1αiαjP(Ai∩Aj) ≥
0.
2 Balls and Bins
Throw n balls into m bins, where m and n are positive integers. Let X be the number of bins with
exactly one ball. Compute varX.
3 Portfolio Optimization
Suppose that there are n assets, where n is a positive integer. For each unit dollar invested in asset
i, for i = 1,...,n, with probability pi
the value of the asset will grow by αi
to 1 + αi
, and with
probability 1 − pi
the value of the asset will shrink by αi
to 1 − αi
. Let the proportion of money
invested in asset i be wi (so that ∑
n
i=1wi = 1), and let Xi be a random variable denoting the final
CS 70, Fall 2017, HW 11 1
value of the ith asset per unit dollar. Then X = w1X1 +···+wnXn is the total value. For simplicity,
assume that the outcomes of the different assets are independent.
(a) Compute the expectation E[X]. What values of wi maximize this quantity?
(b) Compute the variance varX. What values of wi minimize this quantity?
4 Uniform Means
Let X1,X2,...,Xn be n independent and identically distributed uniform random variables on the
interval [0,1] (where n is a positive integer).
(a) Let Y = min{X1,X2,...,Xn}. Find E(Y). [Hint: Use the tail sum formula, which says the
expected value of a nonnegative random variable is E(X) = R ∞
0 P(X x)dx. Note that we can
use the tail sum formula since Y ≥ 0.]
(b) Let Z = max{X1,X2,...,Xn}. Find E(Z). [Hint: Find the CDF.]
5 Darts (Again!)
Alvin is playing darts. His aim follows an exponential distribution; that is, the probability density
that the dart is x distance from the center is fX(x) = exp(−x). The board’s radius is 4 units.
(a) What is the probability the dart will stay within the board?
(b) Say you know Alvin made it on the board. What is the probability he is within 1 unit from the
center?
(c) If Alvin is within 1 unit from the center, he scores 4 points, if he is within 2 units, he scores
3, etc. In other words, Alvin scores b5−xc, where x is the distance from the center. What is
Alvin’s expected score after one throw?
6 Exponential Distributions: Lightbulbs
A brand new lightbulb has just been installed in our classroom, and you know the life span of a
lightbulb is exponentially distributed with a mean of 50 days.
(a) Suppose an electrician is scheduled to check on the lightbulb in 30 days and replace it if it is
broken. What is the probability that the electrician will find the bulb broken?
(b) Suppose the electrician finds the bulb broken and replaces it with a new one. What is the
probablity that the new bulb will last at least 30 days?
(c) Suppose the electrician finds the bulb in working condition and leaves. What is the probability
that the bulb will last at least another 30 days?
CS 70, Fall 2017, HW 11 2