Assume that the proportion of voters who prefer Candidate A is p = 0.617 . Organization D conducts a poll of n = 5 voters. Let X represent the number of voters polled who prefer Candidate A. Use some form of appropriate technology (e.g., your calculator or statistics software like Excel, R, or StatDisk) to find the cumulative probability distribution.

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Solution to the problem

Let X the random variable of interest "number of voters polled who prefer Candidate A", on this case we now that:

[tex]X \sim Binom(n=5, p=0.617)[/tex]

The probability mass function for the Binomial distribution is given as:

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]

Where (nCx) means combinatory and it's given by this formula:

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]

We can create the cumulative distribution function with the following R code:

> x<-seq(0,5,1)

> y<-dbinom(x,5,p=0.617)

> plot(x,y,type = "h", main="PMF function") # the pmf function

> x= rbinom(n=10000,5, 0.617)

> P= ecdf(x)

> plot(P) # The cdf function