Probability Distributions: a review Focus on Poisson and Exponential Distributions

Studying the patterns… in a stochastic way!

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

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

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

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

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

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

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

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

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

http://www.boost.org/doc/libs/1_35_0/libs/math/doc/sf_and_dist/graphs/poisson.png

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

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

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

http://users.encs.concordia.ca/~jerry/notes/image129.gif

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

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

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

Exponential Distribution http://www.efunda.com/math/distributions/images/ExpDistPlot.gif

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

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