Michael Lugo
  • Member for 11 years, 6 months
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What do $\pi$ and $e$ stand for in the normal distribution formula?
Accepted answer
90 votes

So I think you want to know "why" $\pi$ and $e$ appear here based on an explanation that goes back to circles and natural logarithms, which are the usual contexts in which one first sees these. If ...

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Why does the first 100,000 zeroes of the Riemann Zeta function have double-digit sequence count discontinuities at 00,11,22,33,44,55,66,77,88,99?
66 votes

This has nothing to do with the Riemann zeta function, but is rather a property of random sequences of digits. (As a rule there's no reason to assume that there's anything significant about the ...

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Find the average of $\sin^{100} (x)$ in 5 minutes?
60 votes

Assuming Arnold means to find an approximate value -- I'd do this as follows: first, we may as well find the average of $\cos^{100} x$. I'll do this over a half-period, $-\pi/2 \le x \le \pi/2$. But $\...

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Proving that $1$- and $2D$ simple symmetric random walks return to the origin with probability $1$
47 votes

See Durrett, Probability: Theory and Examples (link goes to online copy of the fourth edition; original defunct link). On p. 164 Durrett gives a proof that simple random walk is recurrent in two ...

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Probability of 3 people in a room of 30 having the same birthday
37 votes

An exact formula can be found in Anirban DasGupta, The matching, birthday and the strong birthday problem: a contemporary review, Journal of Statistical Planning and Inference 130 (2005), 377-389. ...

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Can you answer my son's fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?
36 votes

From Srivatsan Narayanan's comment: there are on the order of $n^7$ numbers satisfying the digit constraint, with $n$ digits. The probability that a random $n$-digit number is prime is of order $1/n$. ...

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How are we able to calculate specific numbers in the Fibonacci Sequence?
33 votes

A lot of people have mentioned Binet's formula. But I suspect this is not the most practical way to compute the nth Fibonacci number for large n, because it requires either having a very accurate ...

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If a prime number is reversed, and then appended to itself, why is the result always a composite number?
27 votes

As has been pointed out in the comments, this is not a special property of the primes. Rather it's true that whenever you reverse a number and append the result to itself, the result is always ...

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Is there a real number lookup algorithm or service?
25 votes

I've long used Simon Plouffe's inverse symbolic calculator for this purpose. It is essentially a searchable list of "interesting" numbers.

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Math story: Ten marriage candidates and 'greatest of all time'
24 votes

This is the secretary problem. If there are $n$ suitors, the optimal strategy is to reject the first $r-1$ of them and then accept the first one that is better than all of those $r-1$. In general $r \...

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Can I use my powers for good?
24 votes

Cathy O'Neil gave a talk at MIT entitled "Math in Business" last week; she summarizes that talk in this blog post. There may be some ideas here.

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Why is $1^{\infty}$ considered to be an indeterminate form
23 votes

Look at the logarithm. More specifically, consider $f(x)^{g(x)}$ as $x \to \infty$, where $\lim_{x \to \infty} g(x) = \infty$ and $\lim_{x \to \infty} f(x) = 1$. (This is something of form $1^\infty$....

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Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$
20 votes

A probabilistic method that I learned from Jim Pitman's book Probability (exercise 3.3.10) is as follows. Let $X$ be uniformly distributed on the set $\{ 1, 2, \ldots, n \}$. Then $$ E(X^3) = (1^3 + 2^...

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$C(n,p)$: even or odd?
19 votes

Yes. We only need to know how many powers of two appear in the prime factorization of C(n,p). The number of powers of two that appear in the prime factorization of $n!$ is $\lfloor n/2 \rfloor + \...

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Why do the $n \times n$ non-singular matrices form an "open" set?
19 votes

Somewhat informally: if you take a matrix, and change the entries of it a little bit, then the determinant also changes by a little bit. (Formally, the determinant is a continuous function of the ...

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Good Physical Demonstrations of Abstract Mathematics
19 votes

The book "The Mathematical Mechanic" by Mark Levi is a very good source of such examples, which Levi has been collecting for some time. The first two here are in the book, if I recall correctly.

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When you randomly shuffle a deck of cards, what is the probability that it is a unique permutation never before configured?
Accepted answer
18 votes

Your original answer of $\dfrac{3 \times 10^{14}}{52!}$ is not far from being right. That is in fact the expected number of times any ordering of the cards has occurred. The probability that any ...

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Why eliminate radicals in the denominator? [rationalizing the denominator]
18 votes

The usual reason I've heard is that dividing by integers is computationally easier -- it's easier to find, say, $(5\sqrt{3})/3$ by computing $5 \times \sqrt{3} \approx 8.66 $ and then dividing by $3$ ...

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A (probably trivial) induction problem: $\sum_2^nk^{-2}\lt1$
17 votes

Another proof is by comparison: note that $$ {1 \over k^2} < {1 \over (k-1)k} $$ for all integers $k \ge 2$. Therefore $$ {1 \over 2^2} + {1 \over 3^2} + \cdots + {1 \over n^2} < {1 \over 1 ...

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Chance of meeting in a bar
17 votes

I'd say that each of them arrives at some time uniformly distributed on [12:00, 12:45], where the times are independent. They meet if their arrival times differ by less than fifteen minutes. In the ...

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Factoring $a^{10}+a^5+1$
16 votes

Another hint: can you factor $a^{15}-1$? In more than one way? Six years later, for the record, let's spell out what I meant by this. What I had intended was to observe $$a^{15}-1 = (a^3)^5 - 1 = (...

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Show that any two consecutive odd integers are relatively prime
16 votes

If you know about the Euclidean algorithm, you can see that $\gcd(2k+3, 2k+1) = \gcd(2k+1, 2) = \gcd(2, 1) = 1$. As for making a linear combination: try small values of $k$ instead of diving right ...

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what is the sum of following permutation series $nP0 + nP1 + nP2 +\cdots+ nPn$?
16 votes

You can write this as $$ S(n) = n! \left( {1 \over 0!} + {1 \over 1!} + \cdots + {1 \over n!} \right) $$ and now recall that $e = 1/0! + 1/1! + 1/2! + \cdots $. So in fact $$ S(n) = n! \left( e - \...

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$m!n! < (m+n)!$ Proof?
16 votes

One-line proof (some details omitted): ${m+n \choose m} > 1$ if $0 < m < n$.

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Uniform distribution on a simplex via i.i.d. random variables
Accepted answer
16 votes

Take a look at the Wikipedia article on the Dirichlet distribution. In particular the Dirichlet distribution with $\alpha_i = 1$ for all $i$ is the uniform distribution on the simplex. Furthermore, ...

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Rolling $2$ dice: NOT using $36$ as the base?
15 votes

Paint the dice different colors, say red and blue. Now (red 1, blue 2) is clearly different from (red 2, blue 1). But the dice don't know they're painted, because they're dice.

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$x^y = y^x$ for integers $x$ and $y$
15 votes

Say $x^y = y^x$, and $x > y > 0$. Taking logs, $y \log x = x \log y$; rearranging, $(\log x)/x = (\log y)/y$. Let $f(x) = (\log x)/x$; then this is $f(x) = f(y)$. Now, $f^\prime(x) = (1-\log x)/...

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Taking Seats on a Plane
15 votes

I don't really have the intuition for this, but I know the formal proof. This is equivalent to showing that the probability that in a permutation of $[n]$ chosen uniformly at random, two elements ...

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Parenthesis vs brackets for matrices
15 votes

They're interchangeable. I think a lot of people tend to use parentheses just because they're easier to write by hand.

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Does sum of the reciprocals of all the composite numbers converge?
14 votes

This sum does not converge. Consider for example the sum of the reciprocals of all the even numbers greater than 2: $1/4 + 1/6 + 1/8 + 1/10 + \ldots$. This diverges, the proof being analogous to the ...

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