# Poisson distribution

### Poisson distribution

#### Lessons

The Poisson Distribution is an approximation to the Binomial Distribution.

Recall:
$P(x)= {_n}C_{x}\;p^x(1-p)^{n-x}$
$n$: number of trials
$x$: number of success in n trials
$p$: probability of success in each trial
$P(x)$: probability of getting x successes (out of n trials)
$\mu=np$
Now:
$\mu=\lambda=np$

Poisson Distribution:
$P(x)=e^{-\lambda}$$\frac{\lambda^x}{x!}$
• poissonpdf $(\lambda,x)$
• poissoncdf $(\lambda,x)$
• Introduction
What is the Poisson Distribution?

• 1.
Determining the Poisson Distribution
The number of meteors that hit the earth in a given day is modelled by a Poisson Distribution with $\lambda=4$. What is the probability that 5 meteors hit the earth in a day?

• 2.
When making a video I typically make 1 error for every 20 minutes of video time. If I make 45 minutes of video what is the probability that I make 3 errors?

• 3.
In a particular community the average person survives to age 100 with probability 0.005 (which is equivalent to 0.5%). If this community has 2,000 people, then what is the probability that 15 people in this community survive to age 100 using;
a)
The Binomial Distribution

b)
The Poisson Distribution

c)

• 4.
Cumulative Poisson Distribution
On U.S. route 66 (an American highway) every car that travels this whole route has a probability of $p=0.0002$ of getting into a car accident. A total of 10,000 cars drive this route every month. What is the probability that there are fewer than 3 car accidents in a month?

• 5.
Determining the Poisson Distribution using Calculator Commands
A fair coin is flipped 10 times, what is the probability using the Poisson Distribution commands on your calculator find,
a)
The probability that heads comes up 5 times?

b)
The probability that heads comes up 5 or less times?

c)
The probability that heads come up more than 7 times?