R – How to Generate a Random Integer

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I am trying to generate a random integer that follows this distribution.

$$ P(X=k)=p(1-p)^{k-1}$$
for
$$ k=1,2,…$$

I tried to use sample with some numbers, but i do not know how to use the distribution.

Best Answer

This is the probability mass function of the geometric distribution.

If you use R: To sample from the geometric distribution, first specify the probability $p$ and then use the rgeom function:

# The geometric distribution has one parameter, a probability between 0 and 1.
p <- 0.3
# How many numbers do you want to generate?
n <- 10
# Draw random sample from the geometric distribution.
rgeom(n, p)
#>  [1] 7 0 6 8 0 1 0 0 2 2

Related Solutions

Poisson Distribution – Using Poisson Distribution to Generate Random Integers

I'm trying to generate random integers which have Poisson distribution.
I want to know what it actually returns. Does it return k for a particular value of p(k)?

No. The part you quoted describes what it does quite clearly. It returns a random $X$ which has $P(X=k)=p(k)$. If you tell it $\mu$ you'll get a set of random counts from a Poisson($\mu$) distribution.

For a particular value of $\mu$, $k$ can take values from 0,1,2,3...
So if I specify a value for $\mu$, do I have an upper bound on the values returned by this function?

No. For any fixed value $M$, there's still some chance of exceeding $M$, though it may be vanishingly small. (Well, because it's implemented on a computer with finite precision, there's always some limit - you can't store something bigger than the biggest integer, for example - and the exact behavior as it approaches or goes beyond whatever limit exists depend on implementation.)

Random Numbers – Generate Random Numbers Following a Distribution Within an Interval

It sounds like you want to simulate from a truncated distribution, and in your specific example, a truncated normal.

There are a variety of methods for doing so, some simple, some relatively efficient.

I'll illustrate some approaches on your normal example.

Here's one very simple method for generating one at a time (in some kind of pseudocode):

$\tt{repeat}$ generate $x_i$ from N(mean,sd) $\tt{until}$ lower $\leq x_i\leq$ upper

If most of the distribution is within the bounds, this is pretty reasonable but it can get quite slow if you nearly always generate outside the limits.

In R you could avoid the one-at-a-time loop by computing the area within the bounds and generate enough values that you could be almost certain that after throwing out the values outside the bounds you still had as many values as needed.
You could use accept-reject with some suitable majorizing function over the interval (in some cases uniform will be good enough). If the limits were reasonably narrow relative to the s.d. but you weren't far into the tail, a uniform majorizing would work okay with the normal, for example.
If you have a reasonably efficient cdf and inverse cdf (such as pnorm and qnorm for the normal distribution in R) you can use the inverse-cdf method described in the first paragraph of the simulating section of the Wikipedia page on the truncated normal. [In effect this is the same as taking a truncated uniform (truncated at the required quantiles, which actually requires no rejections at all, since that's just another uniform) and apply the inverse normal cdf to that. Note that this can fail if you're far into the tail]
There are other approaches; the same Wikipedia page mentions adapting the ziggurat method, that should work for a variety of distributions.

The same Wikipedia link mentions two specific packages (both on CRAN) with functions for generating truncated normals:

The MSM package in R has a function, rtnorm, that calculates draws from a truncated normal. The truncnorm package in R also has functions to draw from a truncated normal.

Looking around, a lot of this is covered in answers on other questions (but not exactly duplicates since this question is more general than just the truncated normal) ... see additional discussion in

a. This answer

b. Xi'an's answer here, which has a link to his arXiv paper (along with some other worthwhile responses).

Best Answer

Related Solutions

Poisson Distribution – Using Poisson Distribution to Generate Random Integers

Random Numbers – Generate Random Numbers Following a Distribution Within an Interval

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