Answer to Question #278342 in Statistics and Probability for Henry

Question

Vocabulary and R functions

We are now covering material that is not common knowledge, and it is now very important to include citations when you use an idea from another source (book, Internet, etc). Be sure to add a citation and reference entry for the definition questions below.

a) What is an exponential distribution (include an APA citation)? ___________________________

b) When would you use an exponential distribution? ________________________

c) What is a binomial distribution (include an APA citation)?_______________________

d) When would you use a binomial distribution? ___________________

Answer 1

a)

The exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

Probability density function:

$f(x,\lambda)=\begin{cases} \lambda e^{-\lambda x} &\text{ } x\ge 0 \\ 0 &\text{} x<0 \end{cases}$

where $\lambda>0$ is the parameter of the distribution, called the rate parameter.

b)

Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts.

c)

The binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p).

The probability of getting exactly k successes in n independent Bernoulli trials is given by the probability mass function:

$f(k,n,p)=\begin{pmatrix} n \\ k \end{pmatrix} p^k(1-p)^{n-k}$

d)

We can use the binomial distribution to find the probability of getting a certain number of successes, like successful basketball shots, out of a fixed number of trials. We use the binomial distribution to find discrete probabilities.

References:

https://en.wikipedia.org/wiki/Exponential_distribution

https://en.wikipedia.org/wiki/Binomial_distribution