Lab2: Discrete Distributions

Getting Started with R

Click on the R icon

If no R icon, go to Start and then All Programs to find R

The Binomial Distribution

(some from Dr. Geyer's web page)

Every distribution that R handles has four functions. There is a root name, for example, the root name for the normal distribution is norm. This root is prefixed by one of the letters

p for "probability", the cumulative distribution function (c. d. f.)
q for "quantile", the inverse c. d. f.
d for "density", the density function (p. f. or p. d. f.)
r for "random", a random variable having the specified distribution

For the binomial distribution, these functions are pbinom, qbinom, dbinom, and rbinom.

For help, use the help function on one of the four functions above.

What arguments are needed to specify the parameters?

pbinom is the R function that calculates the c.d.f. of the binomial distribution.

Example: Return to Dr. Dribble (Exercise 3.1-2)

Dr. Dribble has a probability of .8 of making free throws each time he shoots.
Assume his shots are independent of each other.

Let X be the number of free throws made. X is has a Binomial(10, 0.8) distribution. What does this look like?

Let's generate a large number of Binomial(n,p) random variables and look at the histogram.

> set.seed(1)
> bindat <- rbinom(n=10000, size=10, prob=0.8)
> hist(bindat, breaks=seq(2, 10, 1), freq=F)

We want to find the probability of making at least 8 out of 10 free throws.

Let's calculate P(X <= 7) when X is has the Binomial(10, 0.8) distribution using the pbinom function.

> pbinom(7, 10, 0.8)

Now, to answer the question: P(X >= 8) = 1 - P(X <= 7) = 1-0.3222005 = 0.6777995

How does this answer compare to the answer using Table 2 Appendix A?

Example

Question: Suppose widgits produced at Acme Widgit Works have probability 0.005 of being defective. Suppose widgits are shipped in cartons containing 25 widgits. What is the probability that a randomly chosen carton contains no more than one defective widgit?

Question Rephrased: What is P(X <= 1) when X has the Bin(25, 0.005) distribution?

Example: Using loops to simulate a coin toss

(For this example Heads is a success or 1)

for

for() loops are written in the form:

               for( index in values) { expressions }

where the curly brackets are optional when only one expression is specified.

for() loops may also be used to construct data objects by using index as a subscript for the variable being created. In this case, the variable must be initiated outside the for() loop.

The program is:

nreps <- 10
temp <- rep(NA,nreps)
set.seed(5)
for (i in 1:nreps){
  temp[i] <- sample(c(1,0), size=1, replace=T)
}

We actually do not need loops to perform 10 Bernoulli trials, we can

set.seed(5) 
temp2 <- sample(c(1,0), size=10, replace=T)

How would you simulate a coin toss when the probability of success is 0.25?
Hint: Look at the help file for sample!

Now write a loop that simulates Dr. Dribbles free throws and Y is the number of successes for n=10, 20, 50, 100, and 500 free throw attempts.
Use the sample function.

Now use a R binom function!

Here is the code and output for 2 different loops in drdribble_sim.r

Random Number Generators

Here is a nice introduction to Random Number Generators

What is the seed in the above introduction?