/sci/ - Two problems that were on my job application

Name: Anonymous 2009-07-24 22:47

After a night of wild revelry, a group of local intoxicated hoodlums stumble into a children's playground. One of these hoodlums, Phil, climbs to the middle of a spin-around carousel and his friends push the carousel so it rotates once every five seconds. Phil, who is trying not to get sick, holds a flashlight motionless in his hand.

There is a straight path running by the playground that, at its closest point, is 30 meters from the middle of the carousel. Unknown to Phil, there are two cops facing him on the path, shocked at the spectacle. One of them is standing on the path at the point closest to the carousel, while the other is standing 60 meters down the path. At approximately what speed (in meters per second) does the spot illuminated by the flashlight traverse each of the cops' bodies? State speed at the closest cop first.

A rustic village contains one million married couples and no children. Each couple has exactly one child per year. Each couple wants a girl, but also wants to minimize the number of children they have, so they will continue to have children until they have their first girl. Assume that children are equally likely to be born male or female. Let p(t) be the percentage of children that are female at the end of year t. What is p(t)? "Can't tell" is a potential answer if you don't have sufficient information.

Let's see how fast you guys can answer these.

Name: Anonymous 2009-07-30 0:49

>>17
Not bad, actually, except for the \binom thing.

Continued:

OK, so that's fucking awful, so we'll use the central limit theorem instead. Y will then be normal with mean equal to 1000000 * (mean of Y_i) and variance = 1000000 * (variance of Y_i). To get the mean and variance of Y_i we do

E(Y_i) = 1 \cdot \frac{1}{2} + 2 \cdot \left(\frac{1}{2}\right)^2 + ... + (t-1) \cdot \left(\frac{1}{2}\right)^{t-1} + t \cdot \left(\frac{1}{2}\right)^{t-1} = 2-\frac{1}{2^{t-1}}

E(Y_i)^2 = 1^2 \cdot \frac{1}{2} + 2^2 \cdot \left(\frac{1}{2}\right)^2 + ... + (t-1)^2 \cdot \left(\frac{1}{2}\right)^{t-1} + t^2 \cdot \left(\frac{1}{2}\right)^{t-1} = 6-\frac{3}{2^{t-1}}-\frac{t}{2^{t-2}}

Var(Y_i) = 2^{1-2 u} \left(-2+2^u+2^{2 u}-2^{1+u} u\right)

(using mathematica for all the painful sums). Therefore Y has a normal distribution with parameters set to what they should be based on that mess (O_o).

So, we want to find the density function (or something) of X/Y, i.e. what is P(X/Y<n) = P(X<n*Y) given a percentage n. Since we know X <= 1000000, we can write this as a finite sum

\sum_{x=0}{1000000} P(X = x) P(Y > X/n)

and since we know the distributions of X and Y, we can evaluate this.

Two problems that were on my job application

1 Name: Anonymous 2009-07-24 22:47

18 Name: Anonymous 2009-07-30 0:49

Name: Anonymous 2009-07-24 22:47

Name: Anonymous 2009-07-30 0:49