| Abdullah Al Mahmud | www.statmania.info |

Used in situation where some events occur at certain place or time.

- No. of car accidents in a month
- No. of sixes in an innings
- No. of calls in an hour in a call center
- No. of bacteria in a \(mm^2\)
- No. of defective bolts in a lot (in statistical quality control)
- No. of particles in a radioactive decay
- No. of rats in cultivable land
- No. of meteorites greater than 1 meter diameter that strike Earth in a year
- No. of patients arriving in an emergency room between 10 and 11 pm
- No. of laser photons hitting a detector in a particular time interval

\(\displaystyle P(x) = \frac{e^{-\lambda}\lambda^x}{x!}\)

- \(e=2.718\) (constant)
- x: number of occurrences (success)
- \(\lambda\): average no. of events

A process involving ..

- Occurrences are independent
- \(p \space \propto\)Interval
- another

- Mean and variance are same (\(\lambda\))
- mgf: \(e^\lambda e(^t-1)\)
- If \(x_1 \sim Poisson(\lambda_1)\) & \(x_2 \sim Poisson(\lambda_1)\), then \((x_1+x_2) \sim Poisson(\lambda_1+\lambda_2)\)
- skewness: \(\frac 1 \lambda\) (-ve skew)
- kurtosis: platykurtic (\(1+\frac 1 {\sqrt{\lambda}}\))

\(\displaystyle \sum_{i=1}^{\infty} \frac{e^{-\lambda}\lambda^x}{x!}=1\)

Prove

P(x+1)

If \(\lambda\) is very large.

- Number of trials, n, is very large: \(n \to \infty\)
- Probability of success, p, is very low: \(p \to 0\)
- Mean, \(np=\lambda\) finite

- Events that can happen a very large number of time, but happen rarely.
- That is, they are used in situations that would be more properly represented by a Binomial distribution with a very large n and small p, especially when the exact values of n and p are unknown.

- Binomial counts discrete occurrences among discrete trials (finite attempts)
- Poisson counts discrete occurrences among continuous trials (infinite attempts)
- In Poisson, success can occur at any point of time (or space)
- Accidents in road (anywhere anytime), knots on a rope

- Both measure the number of certain random events (or “successes”) within a certain frame - Binomial is based on discrete events, while the Poisson is based on continuous events.
- In a Binomial distribution you have a certain number, n, of “attempts,” each of which has probability of success p.
- With a Poisson distribution, you essentially have infinite attempts, with infinitesimal chance of success.
- Given a Binomial distribution with some n,p, if you let n→∞ and p→0 in such a way that np→λ, then that distribution approaches a Poisson distribution with parameter λ.

If P(x = 2) = P(x = 3), find

- parameters
- \(P(x>0)\)
- \(P(x\le 2)\)
- \(P(x\ge 2)\)

Standard deviation of a Poisson distribution is 4. Find mean and the probabilities in problem 01.

If \(\frac{k\mu^x}{x!}; x = 0, 1, 2, \cdots, \infty,\)

k=?

Overflow floods occur once every 100 years on average. Calculate the probability of k = 0, 1, 2, 3, 4, 5, or 6 overflow floods in a 100-year interval, assuming the Poisson model is appropriate.

- Because the average event rate is one overflow flood per 100 years, λ = 1

Ugarte and colleagues report that the average number of goals in a World Cup soccer match is approximately 2.5 and the Poisson model is appropriate. Estimate probability of k goals and then k = 0,1,2,3..