1. Probability Distributions: a review
Focus on Poisson and Exponential Distributions

2. Studying the patterns… in a stochastic way!

3. Recall Binomial Distribution
Used to calculate the total probability of obtaining x successes from n trials. Probability of success is p.
Discrete

4. Recall Geometric Distribution
Used to calculate the probability that x trials are required in order to obtain the first success (or probability of obtaining x-1 failures followed by one success).
Discrete

5. Recall Poisson Distribution
Similar to the binomial distribution, this distribution can be used to calculate the probability of obtaining x successes (events) but for situations occurring in a continuum.
Discrete

6. Recall Poisson Distribution
Example 1: Number of telephone calls received in a given time interval. Number of telephone calls: discrete Time interval: continuum
“events following a Poisson distribution are said to occur randomly in time”
Discrete

7. Recall Poisson Distribution
Example 2: Number of stars above certain brightness in a particular area of the sky Number of stars: discrete Area of the sky: continuum
Discrete

8. Recall Poisson Distribution
This distribution is the limit of the binomial distribution when the number of trials n approaches infinity and the probability of success p approaches 0 BUT np remains finite.
Discrete

9. Recall Poisson Distribution
In a sufficiently small time interval, at most one event (0 or 1) can occur.
Discrete

10. Recall Poisson Distribution
Actually, 𝒏𝒑=𝝀 which is the mean (average) number of events in a Poisson distribution. 𝝀 is also the variance.
Discrete

11. Recall Poisson Distribution
The probability of obtaining exactly x successes in the given time interval is (the pf): 𝒇 𝒙 =𝑷𝒓 𝑿=𝒙 = 𝒆 −𝝀 𝝀 𝒙 𝒙!
Discrete

13. Recall Poisson Distribution
Problem 1: A person receives on average 3 messages per half-hour interval. Assuming that the s are received randomly in time, find the probabilities that in any particular hour 2 messages are received. Let X be the number of s per hour. 𝑷𝒓 𝑿=𝟐 = 𝒆 −𝟔 𝟔 𝟐 𝟐!
Discrete

14. Recall Exponential Distribution
This occurs naturally if we consider the distribution of the length of intervals between successive events in a Poisson process.
Continuous

15. Recall Exponential Distribution
Or we can interpret it as the distribution of the interval (waiting time) before the first event.
Forgetfulness (memorylessness) property!
Continuous

17. Recall Exponential Distribution
Forgetfulness (memorylessness) property: 𝑷𝒓 𝑻>𝒕+𝒔 𝑻>𝒔}=𝑷𝒓{𝑻>𝒕} Where T is exponentially distributed and s is the interval since the occurrence of the last Poisson event.
Continuous

18. Recall Exponential Distribution
Recall “events following a Poisson distribution are said to occur randomly in time”. Randomness means that the occurrence of an event is not influenced by the length of time that has elapsed since the occurrence of the last event. The exponential distribution is completely random!
Continuous

19. Recall Exponential Distribution
For a positive parameter 𝝀:
𝒇 𝒙 = 𝝀 𝒆 −𝝀𝒙 𝒇𝒐𝒓 𝒙>𝟎 𝟎 𝒇𝒐𝒓 𝒙≤𝟎
The first event occurs in interval [x,x+dx].
Mean = 𝟏 𝝀 Variance = 𝟏 𝝀 𝟐
Continuous

20. Exponential Distribution

21. Recall Gamma Distribution of order r
This is a generalization of the exponential distribution (r=1). This considers the distribution of the interval between every r-th event in a Poisson process (waiting time before the r-th success).
Continuous

22. Remarks
For large r, the Gamma distribution tends to the Gaussian (Normal) distribution. The Gaussian (Normal) distribution can also be used to approximate the Poisson distribution when the mean 𝝀 becomes large (e.g., 𝝀≥𝟏𝟎).
Continuous