Interview Questions Reloaded

Khoa Tran · 10-18-2007, 09:47 AM

Có lẽ sau chiêu sư tử hống $n\sin(n)$ (link) của Kim Mao Sư Vương Ngô Quang Hưng, quần hùng lăn ra giãy đành đạch, không còn ai hứng thú thi triển võ công nữa. Thôi thì chúng ta lại viết tập mới với mấy câu hỏi dễ thương hơn.

1. Let $W_t$ be a Brownian motion. What is $\mathbb{E}[W_t^6]$ ?

2. How do you compute $\int_0^\infty e^{-3x^2}dx$

3. (from YADD) After each second, an insect may: either die, or survive, or split into 2, or split into 3 (each with probability 0.25). Suppose that there's 1 insect at the beginning of the day, what is the probability that the whole population dies at the end of the day?

4. (Programming) Given 2 variables a and b. How do you swap the values of a and b without using any extra memory?

5. (Blog KHMT) Giả sử

(mũ x đến vô cùng). Vậy x bằng bao nhiêu? Ví dụ, ta có thể lý luận như sau. Dễ thấy rằng

. Do đó

, hay

. Nhưng nếu

thì, theo lý luận trên,

cũng dẫn đến

. Sao lạ thế?

à?

YADD · 10-19-2007, 06:49 AM

1) A cute trick is compute

$\int_0^\infty e^{-a\frac{W^2}{2}}dW$

then take drivative w.r.t. a thrice.

3) It is a bacterium, Khoa. An insect cannot duplicate that fast. The tougher part of the puzzle is justify the solution (i.e., discarding the probability = 100% answer). A clue is realted to puzle 5) below.

5) Let me repharse the question. Define $y = x^{x^{x^{x^{...}}}}$ (infinite times). For $x = \sqrt{2}$ , find y.

The paradox comes about like this: Rewrite $y = x^y$ ; then with $x=\sqrt{2}$ , both $y=2$ and $y=4$ satisfy this equation.

The paradox is resolved as follows: Define $y_n = x^{x^{x^{x^{...}}}}$ (n times), equivalently $y_{n+1} = x^{y_n}$ . Plot $y_{n+1}$ against $y_n$ on a two-dimensional axes, the curve (for $x=\sqrt 2$ ) is a monotonically increasing function crossing the 45-degree line at two points A (y=2) and B (y=4). The slope of the curve at A is smaller than unity [one], so A represents an attractive (stable) fixed point. The slope at B point is greater than unity => B is a repulsive (unstable) fixed point. So $y_\infty \rightarrow 2,$ while point B (y=4) is not attainable.

[y=4 is a spurious root since some "non-linearity" was introduced in rewriting the definition of y.]

An aside observation from the resolution: The maximum value of x such that $y = x^{x^{x^{x^{...}}}}$ (infinite times) is well-defined is $x^\star = e^{\frac{1}{e}}$ , corresponding to $y = e$ . Beyond $x^\star$ , y explodes.

shinichi9htv · 10-19-2007, 03:56 PM

4) a = a+b, b = a -b, a = a -b

YADD · 10-21-2007, 11:35 PM

Follow-up with question (5)...

The function $f(x) = x^{x^{x^{x^{...}}}} (\infty\ times)$ is well-defined only in the interval $x \in \[e^{-e},e^{\frac{1}{e}}\]$ . The lower bound $e^{-e}$ comes about when the curve previously described crosses the 45-degree line at $-1$ slope. Below the lower bound, a bifurcation occurs (no chaos).

Back to the original question, interestingly, for $y>e$ , e.g., $y=10$ , although the equation $y=x^y$ yields a root $x=10^{\frac{1}{10}}$ , the root is completely spurious! This value of y is not attainable.

nguyenxuanson · 10-25-2007, 09:29 PM

4/ what happen if a+b is out of range of the data type?

drew · 10-25-2007, 10:03 PM

The answer should be something along the line of:
a = a ^ b
b = a ^ b
a = a ^ b
And oh yes, a hypothetical follow-up question would be "what about signed numbers, ANSI C convention 'n stuff ?", at which point I would stand up, stab the interviewer, hypothetically of course, with my hypothetical pencil, of course

!

drew · 10-25-2007, 10:05 PM

Btw hopefully I'll be able to get back next week with a solution to the n sin(n) question, it shouldn't be that hard !

nguyenxuanson · 11-07-2007, 11:48 PM

in this question, we have a bar and we will break it in three sections by two cuts. What is the probability that the three sections form a triangular:
1/ if we first cut the bar in two sections (the position is uniformly distributed) and then choose the section (normally the longer section) where you will cut it (the position is uniformly distributed).
2/ you cut the bar at two places simultaneously (the positions ảe uniformly distributed).

shinichi9htv · 11-13-2007, 06:39 AM

Có 1 câu mới nhớ ra, hồi xưa bị thằng bạn đố, khó phết.

Có 124 con bài (gộp 2 bộ bài lại) và 2 nhà toán học. 1 người thứ 3 cầm lấy bộ bài, lấy ra 5 con bất kì, đưa cho 1 nhà toán học. Ông này ngồi xem xem tính toàn một thôi một hồi, đút lại 1 con vào bộ bài và xếp 4 con ngay ngắn trên bàn. Nhà toán học kia đứng lên nhìn 4 con bài, rồi tìm ra con bài thứ 5 còn lại. Hỏi ông ta làm thế nào?

nguyenxuanson · 11-13-2007, 06:42 PM

given a Brownian motion start from 0 $W_{t}$ we define the time change
$A(t)=\int_{0}^{t}e^{W_{s}+\nu s} ds$
give one presentation of the variable $W(A(t))$ .

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