Tagged Questions
27
votes
2answers
2k views
Why is the error function defined as it is?
$\newcommand{\erf}{\operatorname{erf}}$
This may be a very naïve question, but here goes.
The error function $\erf$ is defined by
$$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$
Of ...
21
votes
6answers
526 views
Does exceptionalism persist as sample size gets large?
Which of the following is more surprising?
In a group of 100 people, the tallest person is one inch taller than the second tallest person.
In a group of one billion people, the tallest person is one ...
18
votes
4answers
464 views
Example of two dependent random variables that satisfy $E[f(X)f(Y)]=Ef(X)Ef(Y)$ for every $f$
Does anyone have an example of two dependent random variables, that satisfy this relation?
$E[f(X)f(Y)]=E[f(X)]E[f(Y)]$
for every function $f(t)$.
Thanks.
*edit: I still couldn't find an example. I ...
12
votes
4answers
415 views
probability and statistics: Does having little correlation imply independence?
Suppose there are two correlated random variable and having very small correlation coefficient (order of 10-1). Is it valid to approximate it as independent random variables?
12
votes
3answers
672 views
Odds of guessing suit from a deck of cards, with perfect memory
While teaching my daughter why drawing to an inside straight is almost always a bad idea, we stumbled upon what I think is a far more difficult problem:
You have a standard 52-card deck with 4 suits ...
11
votes
5answers
8k views
What is the probability of a coin landing tails 7 times in a row in a series of 150 coin flips?
If you were to flip a coin 150 times, what is the probability that it would land tails 7 times in a row? How about 6 times in a row? Is there some forumula that can calculate this probability?
11
votes
3answers
274 views
Efficient computation of $E\left[\frac{1}{1+\sum_iX_i}\right]$ where $X_i$ is RV with Bernoulli distribution with different probabilities
Suppose we have the random variables $X_1, \ldots, X_n$ that have Bernoulli distributions with the (possibly different) probabilities $p_1, \ldots, p_n$. For example, $X_1$ = 1 with probability $p_1$ ...
11
votes
2answers
541 views
You see a route 14 bus on the moon. What is the most likely number of bus routes on the moon?
This question was asked on a forum and while many argued that the answer is 14 (since the probability of you seeing bus 14 is maximum in this case), I argued against it that they were working ...
10
votes
3answers
615 views
With $n$ balls and $n$ bins, what is the probability that exactly $k$ bins have exactly $1$ ball?
I've got a balls and bins problem. Suppose I throw $n$ balls uniformly at random into $n$ bins. What is the probability that exactly $k$ bins end up with exactly $1$ ball?
I know this seems a ...
10
votes
3answers
145 views
Finding a more direct way to reach $\mathbb{E} \left( \sum (X_i - \mu)^2 \right) - \mathbb{E} \left( \sum (X_i - \overline{X})^2 \right) = \sigma^2$
Let $X_i$ be independent random variables, $\forall\,i \in \mathbf{n} \equiv \{0,\dots,n-1\}$, with identical expectation value $\mathbb{E}(X_i)=\mu$, and identical variance ...
9
votes
1answer
172 views
Expectancy value for the percentage of points lying in the Convex Hull (3D)
Suppose I chose n uniformly distributed random points in a 3D cube. What is the expected value for the percentage of points lying on the convex hull as a function of n?
Just as a reference, I made ...
9
votes
1answer
214 views
Hyper Birthday Paradox?
There are $N$ buckets.
Each second we add one new ball to a random bucket - so at $t=k$, there are a total of $k$ balls collectively in the buckets.
At $t=1$, we expect that at least one bucket ...
7
votes
2answers
185 views
Finding $E\left[\frac{\sum_{i=1}^n X_i^2}{(\sum_{i=1}^n X_i)^2}\right]$ of a sample of gamma random variables
Suppose $X_1,\ldots,X_n$ is a random sample from the $\Gamma(k,\lambda)$ distribution where $\lambda$
is unknown and $k$ is a positive integer and known. How can I find $$E\left[\frac{\sum_{i=1}^n ...
7
votes
3answers
805 views
What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6?
Similar to:
"What is the expected number of dice one needs to roll to get 1,2,3,4,5,6 in order?"
but we allow repeats so 1,1,2,2,3,4,4,4,4,5,5,6 would count.
My answer (or simulation) is flawed as I ...
7
votes
4answers
1k views
Intuition behind using complementary CDF to compute expectation for nonnegative random variables
I've read the proof for why $\int_0^\infty P(X >x)dx=E[X]$ for nonnegative random variables (located here) and understand its mechanics, but I'm having trouble understanding the intuition behind ...
