# Simulation of binomial random variables

I am getting started in R. I need help with a command to:

"Simulate 10000 binom(20,0.3)-RV's". I would do it like:

binsim <- rbinom(10000, 20, 0.3)


Question cont.: "Let X be a binomial(20,0.3)-RV. Use simulated numbers to estimate: P(X<=5) and P(X=5)"

Okay for the first I would use: pbinom(???) and for the second dbinom(?????)

How do I connect the simulation results from binsim and the X-RV's to get the required probabilities?

Could I do it with length(binsim(binsim<=5)/length(binsim)?

• Is this homework? – gung - Reinstate Monica Aug 4 '12 at 15:30
• not really, it is an exercise from the book by Brown and Murdoch, First Course in stat. progr. with R, Doing it also as a possible prep. for an actuarial course. Regards – Andreas Rybicki Aug 7 '12 at 13:31
• gung, Thanks, just did it. Q: how could I store ocram's answer, will it stay in my account as long as I wish? Sorry, Q is serious. Thanks again. – Andreas Rybicki Aug 7 '12 at 16:55
• No problem. CV is intended as both a simple Q&A site and also a dynamic, permanent repository of knowledge. This question will remain listed on your user page under the questions you've asked. You & any future visitor will be able to access & learn from it. – gung - Reinstate Monica Aug 7 '12 at 17:26

Functions like pbinom(), ppois(),... are used to compute the true values of $\Pr(X \leq x)$. Similarly, dbinom(), dpois(),... are used to compute true values of $\Pr(X = x)$. Here you are asked to perform simulations. Simulations are relevant when you do not have closed-form expressions for $\Pr(X \leq x)$ or $\Pr(X = x)$. The basic idea is to draw a sample and to evaluate the empirical version of the quantity of interest. For example, if you have drawn $\{0, 2, 7, 3, 10\}$, the empirical version of $\Pr(X \leq 2)$ is $$\frac{\#\{0, 2\}}{\#\{0, 2, 7, 3, 10\}}= \frac{2}{5}=0.4.$$

We do have closed-form expressions for the binomial random variable but, as pointed out by @gung, this looks like an exercice. So, here is an example to help.

Let $X \sim N(\mu=3, \sigma^2 = 1.3)$. We know that $\textrm{E}(X) = \mu = 3$ and that $\textrm{Var}(X) = \sigma^2 = 1.3$, but let us check that by simulations:

> x <- rnorm(1000, mean=3, sd=sqrt(1.3))
> mean(x)
[1] 3.078271
> var(x)
[1] 1.315806


You might find useful to type help(dbinom)` first...

• ocram, thanks, your answer helped me. Unfortunately, in my stat.-courses I never came across simulations, (as far as I remember). Sorry if you receive three times the same answer. This exercise is from book by J. Brown and D. Murdoch. – Andreas Rybicki Aug 7 '12 at 13:21