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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 $(\lambda,x)$

Recall:

• $P(x)= {_n}C_{x}\;p^x(1-p)^{n-x}$

$n$

$x$

$p$

$P(x)$

• $\mu=np$

Now:

• $\mu=\lambda=np$

Poisson Distribution: $P(x)=e^{-\lambda}$$\frac{\lambda^x}{x!}$

• poissonpdf $(\lambda,x)$

• poissoncdf $(\lambda,x)$

- 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 Distributionb)The Poisson Distribution
- 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?

4.

Discrete Probabilities

4.1

Probability distribution - histogram, mean, variance & standard deviation

4.2

Binomial distribution

4.3

Mean and standard deviation of binomial distribution

4.4

Poisson distribution

4.5

Geometric distribution

4.6

Negative binomial distribution

4.7

Hypergeometric distribution

4.8

Properties of expectation

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