Quantitative Interview questions and answers

[quantyst]

Originally Posted by quantyst

Solution:
Let f(n,k) denote the number of ways to place n mathematicians in k rooms such that in each room there will be at least one mathematician.
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This f(n,k) is the Stirling's number of the second kind and read my post regarding the same for more information(and a solution).

Thanks!

 

[IlyaKEightSix]

My apologies. I had a gut feeling that something was wrong but couldn't figure out the exact reason(I should have spent a moment testing my results). The problem arises due to permutations inside a room.
After spending more time on it(and brushing up my P&C), I finally arrived at something.
The solution uses Stirling's number.
Stirling's Number S(n,k): The number of ways to partition a set A {#(A)=n} into k parts.
For case n=k the answer is trivial. Only interesting solution exists for n > k.
If the rooms were indistinguishable then the answer would be S(n,k).(as from the definition of Stirling number)
Since the rooms are distinguishable, we can permute the rooms and thus the answer to the above problem is k!*S(n,k).
For more on Stirlings number read PlanetMath: Stirling numbers of the second kind
Nice question Quantyst. I want more.

Wait, wouldn't the answer have to be n!*S(n,k) since if you have say 10 mathematicians and 3 rooms, that you can put 1 of 10 in the first, 1 of 9 in the second, 1 of 8 in the third, and then arrange the other 7 in any way?

 

[Jerry]

First A(t) is a periodic function with period 12, any t value which is not between 0 and 12 is converted to t'=mod(t,12) and A(t) = A(t'). Second t=[t]+{t} for t>0, where [t] means the largest integer less than t and {t} means the decimal part of t. We use alpha to measure the angle between the hour hand and the segment of the center of the clock and 12, beta to measure the angle between the minute hand and the segment of the center of the clock and 12.

We also make the t clear, like 5.5 means 5 and .5*60, which is 5:30 and 3.2 means 3 and .2*60, which is 3:12. Then alpha is t/12*2*pi, beta is {t}*2*pi. Since A(t) is the minimal positive angle of alpha-beta, A(t) is within [0,pi]. Thus A(t) is abs((t/12-{t})*2*pi), if this value is within [0,pi], if it is over pi, take A(t) to be 2*pi-abs(t/12-{t})*2*pi.

Letting t represent the time in hours since the most recent 12 midnight, express A(t) as the minimal positive angle in radians between the hour-hand and minute-hand of a regular analog clock. In particular, find A(.25), A(1.08), A(6), A(33), A(100), A(-.25), A(-100).

 

[quantyst]

Do You Double It? When?

From any integer position on the number line, you take a step of integer length k in either direction with probability p(k) where 1<=k<=m, and where m is a positive integer. Let L denote a positive integer greater than m. Starting at position zero, if you reach or surpass L before reaching or surpassing -L, you win $1, otherwise you lose $1. However, at any point in the process you have the one-time-only option of doubling the stakes. (Without concerns about utility) When is it optimal (if ever) to do so?

(You may assume p(k) to be a decreasing sequence, for example, of the form P(k)=a/(2^k) for some appropriate a. At any rate, explicitly state what p(k) is if you choose this route)

 

[quantyst]

Let's Shoot Some Targets!

In the xy-coordinate plane, on the horizontal line y=1 there are infinitely many targets of length L(i), each with endpoints (i, 1) and (i+L(i), 1), where i runs through all integers, positive, negative, and zero. From the position (0,0), a bullet is fired at an angle A (with respect to the positive direction of the x-axis), where A is randomly chosen in the interval [0, pi], with uniform probability. Assume the bullet travels in a straight path. Find the probability that the bullet hits a target for each one of the following cases:

(a) L(i)=1/2, (b) L(i)=1/4, (c) L(i)=3/4, (e) L(i)=1-1/(2^|i|), and (d) in general for L(i)=L, a constant.

 

[ayodayo]

Hello guys,

I have a phone interview for a capital markets quant role....any idea of what kind of questions to expect or what to learn prior to? Thanks.

 

[Andy Nguyen]

Capital markets quant role?
That could be anything. What are the products? Buy side/sell side?

 

[ayodayo]

Its within the global markets risk management group. Not exactly sure what job will comprise of. Are there standard questions that will come up during a phone interview?

 

[quantyst]

Greater Distance

Let d(A,B) denote the Euclidean distance between two points A and B.

In the Euclidean plane there are given a circle and a square that are disjoint but have equal areas. Two points C1 and C2 are randomly chosen in the circle, and two points S1 and S2 are randomly chosen in the square. Find the probability that d(C1,C2) > d(S1,S2).

 

[quantyst]

Greater Angle

Let m(A, B, C) denote the measure of angle ABC.

In the Euclidean plane are given a circle and a square that are disjoint but have equal areas. Three points C1, C2, C3 are randomly chosen in the circle, and three points S1, S2, S3 are randomly chosen in the square. Find the probability that m(C1,C2,C3) > m(S1,S2,S3).

 

[quantyst]

Expectation at Infinity

Define an infinite sequence of random variables X(0), X(1), X(2), ... as follows:

X(0)=1, and for every i>0: X(i) is uniformly distributed over the interval [0, (X(i-1))^a] where a>-1.

Find limit of E[X(i)] as i approaches infinity.

 

[quantyst]

Dice Mania

Consider the following dice game between two players who take turns to roll a number of dice. First off, there is an inexhaustible supply of identical dice. The two players agree in advance on an integer s. The first player rolls a single die. Let Y(1) denote (the number on) the rolled die. If Y(1)>s, the first player wins the game. Otherwise, the second player grabs Y(1) many dice and rolls them. Let Y(2) denote the sum of these Y(1) many rolled dice. If Y(2)>s, then the second player wins the game. Otherwise, the first player grabs Y(2) many dice and rolls them. Let Y(3) denote the sum of these Y(2) many rolled dice. If Y(3)>s, then the first player wins the game. Otherwise, the second player grabs Y(3) many dice and rolls them. So, the game continues in this fashion until one of the two players wins the game, and thus the game ends.


(1)Find the probability that the first player wins the game.

(2)How many turns, on average, does it take before the game ends?

(3)What, on average, is the winning sum Y that the winning player rolls?

(4)Find the answer to question (1) as s approaches infinity.

 

[quantyst]

Identical Twins But Different!

Consider two outwardly identical spherical balls made of the same material, same weight, same color, same surface structure, same in every respect but one: the first ball is a solid whose mass is uniformly distributed throughout the ball, the second has a spherical hollow core with the same center as the ball. The radius of the hollow core is just a bit less than that of the ball itself. The density of the non-hollow mass of the second ball is uniform and greater than the uniform density of the first ball. Assume the balls are real and the specifications are as ideal as humans can make them to be. For example, the surfaces of the two balls are rough enough to cause friction even though the balls are spherical.

Without initially knowing which ball is which, and without subjecting them to any physical deformation, change, or internal probe by the use of x-rays, MRI, or other methods, how can you tell which ball is which by a simple experiment?

 

[thetrystero]

Consider two outwardly identical spherical balls made of the same material, same weight, same color, same surface structure, same in every respect but one: the first ball is a solid whose mass is uniformly distributed throughout the ball, the second has a spherical hollow core with the same center as the ball. The radius of the hollow core is just a bit less than that of the ball itself. The density of the non-hollow mass of the second ball is uniform and greater than the uniform density of the first ball. Assume the balls are real and the specifications are as ideal as humans can make them to be. For example, the surfaces of the two balls are rough enough to cause friction even though the balls are spherical.

Without initially knowing which ball is which, and without subjecting them to any physical deformation, change, or internal probe by the use of x-rays, MRI, or other methods, how can you tell which ball is which by a simple experiment?

put the balls on a smooth surface, and spin them on axis. the one that's more difficult to bring up to speed is the hollow shell (ball 2 has the larger rotational inertia of the two).

 

[quantyst]

That Was Easy, How About This?

Earlier I posted the two spherical balls puzzle, and true to my expectation, it was quickly and easily solved.

Now, here's a new challenge, and as of this writing, I have no solution to it:

Two outwardly identical spherical balls are same in every respect, including both having NO hollow core. But they have a difference: The first of them has a uniformly distributed mass, the second has its mass unevenly distributed but still has the same moment of inertia as the first ball. For example, the second ball's distribution is as follows: both very close to the outer surface and at its core it is more densely distributed than elsewhere so that its angular moment of inertia (or rotational inertia) is the same as that of the first ball. Now, what simple experiment can tell us which ball is which?

 

[IlyaKEightSix]

I would think that you can squeeze them with the same amount of force? And depending on their contraction at a certain time, you can tell which one is which.

 

[quantyst]

I would think that you can squeeze them with the same amount of force? And depending on their contraction at a certain time, you can tell which one is which.

You need to be a bit more circumspect.

This problem comes on the heels of the one preceding it. As before, it goes without saying, you cannot deform them. Can you squeeze steel balls? This is a tough physics problem having to do with angular moments of inertia. A rather (hopefully) simple experiment should determine which is which.

 

[thetrystero]

This is a tough physics problem having to do with angular moments of inertia.

Since the energy and momentum depends solely upon the (spherically symmetric) rotational inertia, I doubt there's a simple kinematic experiment that can be performed to differentiate between them.

quantyst: was this an actual interview question?

 

[quantyst]

Can Eat Cake If You Can Cut It By Two

How do you subdivide a rectangular cake into four equal pieces by two cuts when someone has already sucked out an irregular piece from it? All is 2-D.

Is it even possible to do so? To solve this problem, it would be a good start to prove that it is possible to subdivide what remains of the cake into four pieces of equal area by two straight cuts. A cut may be interpreted as an infinite straight line.

 

[IlyaKEightSix]

How do you subdivide a rectangular cake into four equal pieces by two cuts when someone has already sucked out an irregular piece from it? All is 2-D.

Is it even possible to do so? To solve this problem, it would be a good start to prove that it is possible to subdivide what remains of the cake into four pieces of equal area by two straight cuts. A cut may be interpreted as an infinite straight line.

This may sound naive, but no matter the cut, the cake will still have some form of area. Now divide that area by 4, and partition accordingly by partitioning the irregular cut such that one fourth of it is in each quadrant.