63
votes
Accepted
Why is generating 8 random bits uniform on (0, 255)?
TL;DR:
The sharp contrast between the bits and coins is that in the case of the coins, you're ignoring the order of the outcomes. HHHHTTTT is treated as the same as TTTTHHHH (both have 4 heads and 4 ...
- 92.6k
59
votes
Brain teaser: How to generate 7 integers with equal probability using a biased coin that has a pr(head) = p?
Flip the coin twice. If it lands HH or TT, ignore it and flip it twice again.
Now, the coin has equal probability of coming up ...
- 661
58
votes
Generate a random variable with a defined correlation to an existing variable(s)
I will describe the most general possible solution. Solving the problem in this generality allows us to achieve a remarkably compact software implementation: just two short lines of ...
- 330k
58
votes
Accepted
Generating random numbers manually
If "manually" includes "mechanical" then you have many options available to you. To simulate a Bernoulli variable with probability half, we can toss a coin: $0$ for tails, $1$ for ...
- 23.8k
52
votes
Accepted
Generating correlated binomial random variables
Binomial variables are usually created by summing independent Bernoulli variables. Let's see whether we can start with a pair of correlated Bernoulli variables $(X,Y)$ and do the same thing.
Suppose ...
- 330k
50
votes
Accepted
Finding a way to simulate random numbers for this distribution
There is a straightforward (and if I may add, elegant) solution to this exercise: since $1-F(x)$ appears like a product of two survival distributions:
$$(1-F(x))=\exp\left\{-ax-\frac{b}{p+1}x^{p+1}\...
- 107k
49
votes
Brain teaser: How to generate 7 integers with equal probability using a biased coin that has a pr(head) = p?
Assume that $p \in (0,1)$.
Step 1:. Toss the coin 5 times.
If the outcome is
$(H, H, H, T, T)$, return $1$ and stop.
$(H, H, T, T, H)$, return $2$ and stop.
$(H, T, T, H, H)$, return $3$ and ...
- 47.3k
47
votes
Generating random numbers manually
If you can get access to a very precise clock, you can extract the decimal part of the current time and turn it into a uniform, from which you can derive a normal simulation by the Box-Müller ...
- 107k
45
votes
Accepted
Who is Gail Gasram?
It looks like "Gail Gasram" is "Marsaglia G" (George Marsaglia's surname and first initial) spelled backwards.
- 516
44
votes
Accepted
What exactly is a seed in a random number generator?
Most pseudo-random number generators (PRNGs) are build on algorithms involving some kind of recursive method starting from a base value that is determined by an input called the "seed". The default ...
- 130k
43
votes
Accepted
If I generate a random symmetric matrix, what's the chance it is positive definite?
If your matrices are drawn from standard-normal iid entries, the probability of being positive-definite is approximately $p_N\approx 3^{-N^2/4}$, so for example if $N=5$, the chance is 1/1000, and ...
- 14k
42
votes
Accepted
How to generate random integers between 1 and 4 that have a specific mean?
I agree with X'ian that the problem is under-specified. However, there is an elegant, scalable, efficient, effective, and versatile solution worth considering.
Because the product of the sample mean ...
- 330k
38
votes
Why is the term "Monte Carlo simulation" used instead of "Random simulation"?
Nicholas Metropolis claimed in 1987 that
It was at that
time that I suggested an obvious name
for the statistical method - a suggestion
not unrelated to the fact that Stan[islaw ...
- 129k
33
votes
Accepted
How many numbers can I generate and be 90% sure that there are no duplicates?
The approximation given in the Wikipedia article (also mentioned in the paper by Diaconis & Mosteller (1989)) works well here. Suppose $N$ 4-digit numbers are drawn with replacement from a pool of ...
- 31k
31
votes
Accepted
R: Problem with runif: generated number repeats (more often than expected) after less than 100 000 steps
The documentation of R on random number generation has a few sentences at its end, that confirm your expectation of 32-bit integers being used and might explain what you are observing:
Do not rely ...
- 451
29
votes
Would it be possible to generate data from real data in medical research?
No, this is not good practice.
"Oversampling" the data you have makes your model believe it has more real data than it actually has, and makes it underestimate the probability of actual data ...
- 129k
28
votes
Accepted
How to sample from Cantor distribution?
Easy: sample from a Uniform$(0,1)$ distribution and recode from binary to ternary, interpreting each "1" as a "2". (This is the inverse probability transform approach: it does indeed invert the CDF!)
...
- 330k
26
votes
Accepted
Algorithm for sampling fixed number of samples from a finite population
Yes.
Collect the first $k$ items encountered into the cache. At steps $j=k+1, \ldots, n,$ place item $j$ in the cache with probability $k/j,$ in which case you will remove one of the existing items ...
- 330k
24
votes
Accepted
How do I sample from a discrete (categorical) distribution in log space?
It is possible to sample from categorical distribution given log-probabilities without leaving log space using the Gumbel-max trick. The idea is that if you are given unnormalized log-probabilities $\...
- 140k
23
votes
Brain teaser: How to generate 7 integers with equal probability using a biased coin that has a pr(head) = p?
Generalizing the case described by Dilip Sarwate
Some of the methods described in the other answers use a scheme in which you throw a sequence of $n$ coins in a 'turn' and depending on the result you ...
- 82.8k
22
votes
@whuber 's generation of a random variable with fixed covariance structure
As a point for further elaboration, here is the explanation in the thread you reference:
If it's mathematically possible, [this method] will find an $X_{Y_1,Y_2,\ldots,Y_k;\rho_1,\rho_2,\ldots,\...
- 330k
21
votes
What exactly is a seed in a random number generator?
First, there is no true randomness in today's computer-generated "random numbers." All pseudorandom generators use
deterministic methods. (Possibly, quantum computers will change that.)
The ...
- 57.1k
21
votes
Brain teaser: How to generate 7 integers with equal probability using a biased coin that has a pr(head) = p?
Divide a box into seven equal-area regions, each labeled with an integer. Throw the coin into the box in such a way that it has equal probability of landing in each region.
This is only half in jest -...
- 92.6k
20
votes
Generating Multivariate Uniform Distribution in R
It depends a little bit on the terminology, but usually multivariate uniform refers to a distribution where every point in $[a,b]^d$ is equally likely. Hence, the dimensions are independent, and you ...
- 838
20
votes
Who is Gail Gasram?
Diehard Code
After some extensive digging, it appears that Gail Gasram participated in developing Diehard code, which represented a suite of programs for testing random number generators. Furthermore,...
- 2,280
19
votes
Will this introduce bias into what should be random numbers?
Let's count and see. By construction of the file, all 4-bit strings are equally likely. There are 16 such strings. Here they are:
...
- 20.4k
18
votes
Accepted
What is meant by "Laplace noise"?
You are correct, adding Laplace noise means that to your variable $X$ you add variable $Y$ that follows Laplace distribution. There are multiple reasons why it is called noise. First, think of signal ...
- 140k
18
votes
Accepted
How to generate a $\pm 1$ sequence with mean $0.05$?
Your desired mean is given by equation:
$\frac{N\cdot p - N \cdot (1-p)}{N} = .05$
from which follows that the probability of the 1s should be ...
- 1,196
18
votes
Accepted
Sampling from $x^2\phi(x)$?
Some guesswork suggest that $X$ perhaps can be simulated by a suitable power-transformation of a Gamma random variable $Y$ multiplied by a random sign to make the resulting density symmetric about ...
- 11.5k
17
votes
Why is generating 8 random bits uniform on (0, 255)?
why does a sequence of 8 zeroes or 8 ones seem to be equally as likely as a sequence of 4 and 4, or 5 and 3, etc
The aparent paradox can be summarized in two propositions, that might seem ...
- 1,468
Only top scored, non community-wiki answers of a minimum length are eligible