Definition 5.13.
Discrete Uniform Probability Mass Function:
\begin{equation*}
f(x)=\frac{1}{n}
\end{equation*}
Section Discrete Probability Distributions Worksheet
Subsection Discrete Uniform Probability Mass Function
Subsection The Binomial Distribution
Definition 5.14.
A binomial experiment is a probability experiment possessing the following characteristics:
- It has a fixed number of trials, \(n\text{.}\)
- Each trial has only two possible outcomes: success or failure.
- The probability of success, \(p\text{,}\) and the probability of failure, \(q\text{,}\) are constant.
- Each trial is independent of the other trials.
The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution.
The probability of exactly \(X\) successes in \(n\) trials with probability of success, \(p\text{,}\) is
\begin{equation*}
P(X) = {n\choose X}p^X(1-p)^{n-X}\text{ for } X=0,1,2,...,n
\end{equation*}
where \({n\choose X}=\frac{n!}{X!(n-X)!}\text{.}\)
Good news! The mean and standard deviation of a binomial distribution are simple computations:
\begin{equation*}
\mu = np\text{ and } \sigma=\sqrt{npq}
\end{equation*}
As the number of trials in a binomial experiment increases, calculating the probabilities required using the complex formula becomes tedious. Therefore, we will compute probabilities using Excel formulas.
In Excel:
\(\text{The formula is } BINOM.DIST(x,n,p,\text{cumulative})\text{, where }\text{cumulative=TRUE ( or fewer successes) or FALSE (exactly successes)}\)
\begin{equation*}
\boxed{ P(X\leq x)=BINOM.DIST(x,n,p,TRUE)\;\;\;\text{ and }\;\;\; P(X=x)=BINOM.DIST(x,n,p,FALSE) }
\end{equation*}
Example 5.15.
Subsection The Poisson Distribution
Question 5.16.
We might want to understand how busy traffic is near campus. Let’s say that we stand at the intersection of Clyde Way and Austin Bluffs Parkway and count how many cars drive by on Austin Bluffs from noon-1pm on a particular day. Is this a binomial experiment?
Definition 5.17.
A Poisson process is a probability experiment possessing the following characteristics:
- It counts the number of occurrences of an event over a period of time, area, distance, or some other measurement.
- The mean is the same for each equal interval of measurement.
- The number of occurences in distinct intervals is indpendent.
- The intervals defined in the Poisson process cannot overlap.
Example 5.18.
- \(P(X\lt 5)\)
- \(P(X\gt 5)\)
- \(P(X= 5)\)
- \(P(X\gt 6)\)
- \(P(X\lt 6)\)
- \(P(X= 6)\)
A bank expects to receive six bad checks per day, on average. What is the probability of the bank getting fewer than five bad checks on any given day? Of interest is the number of checks the bank receives in one day, so the time interval of interest is one day. Let \(X=\) the number of bad checks the bank receives in one day. If the bank expects to receive six bad checks per day then the average is six checks per day. Which of the following is a mathematical statement for the probability question.
The outcome for the random variable for a Poisson distribution is the actual number of occurrences of an event over a period of time, area, distance, or any other type of measurement.
The probability of exactly \(X\) occurrences over a given interval is:
\begin{equation*}
P(X)=\frac{\lambda^Xe^{-\lambda}}{X!}\text{, where the mean number of occurrences over the interval.}
\end{equation*}
The mean and standard deviation of a Poisson distribution are even simpler than what we saw for a binomial distribution:
\begin{equation*}
\boxed{ \mu=\lambda }\text{ and }\boxed{\sigma=\sqrt{\lambda}}
\end{equation*}
Once again, we will usually use Excel or tables to compute probabilities associated with Poisson distributions.
(Here is a link to a table that you can use as well:
https://ux1.eiu.edu/~aalvarado2/levine-smume6_topic_POIS.pdf. Make sure to pay attention to if any table you’re using is giving you a cumulative probability or not.)In Excel, we’ll use the formula:
\begin{equation*}
\boxed{POISSON.DIST(x,\lambda,\text{cumulative})}
\end{equation*}
where cumulative=TRUE (\(x\) or fewer occurrences) or FALSE (exactly \(x\) occurrences).
Example 5.19.
(Donnelly 5.23)
Consider a Poisson probability distribution with \(\lambda=5.6\text{.}\) Calculate the probability of exactly 5 occurrences. (Round your answer to 4 decimal places.)
\(P(X=5)=\)
Example 5.20.
(Donnelly 5.23)
Consider a Poisson probability distribution with \(\lambda=5.6\text{.}\) Calculate the probability of more than 6 occurrences. (Round your answer to 4 decimal places.)
\(P(X\gt 6)=\)
Example 5.21.
(Donnelly 5.23)
Consider a Poisson probability distribution with \(\lambda=5.6\text{.}\) Calculate the probability of 3 or fewer occurrences. (Round your answer to 4 decimal places.)
\(P(X\leq 3)=\)
Example 5.22.
(Donnelly 5.29)
A customer support center for a computer manufacturer receives an average of 2.9 phone calls every five minutes. Assume the number of calls received follows the Poisson distribution. Use Excel to answer the following questions.
(a)
What is the probability that no calls will arrive during the next five minutes?
Answer.
\begin{equation*}
P(X=0)=POISSON.DIST(0,2.9,FALSE)\approx 0.0550
\end{equation*}
(b)
What is the probability that 3 or more calls will arrive during the next five minutes?
Answer.
\begin{equation*}
P(X\geq 3)=1-P(X\leq 2)=1-POISSON.DIST(2,2.9,TRUE)\approx 0.5540
\end{equation*}
(c)
What is the probability that 3 calls will arrive during the next ten minutes?
Answer.
\(\lambda=5.8\)
\begin{equation*}
P(X=3)=POISSON.DIST(3,5.8,FALSE)\approx 0.0985
\end{equation*}
(d)
What is the probability that no more than 2 calls will arrive during the next ten minutes?
Answer.
\(\lambda=5.8\)
\begin{equation*}
P(X\leq 2)=POISSON.DIST(2,5.8,TRUE)\approx 0.0715
\end{equation*}
