Multinomial Distribution The Binomial distribution can be extended to describe number of outcomes in a series of independent trials each having more than 2 possible outcomes. If a given trail can result in the k outcomes E1, E2, …, Ek with probabilities p1, p2, …, pk, then the probability distribution of the random variables X1, X2, …, Xk, representing the number of occurrences for E1, E2, …, Ek in n independent trials is with , and STA286 week 5

Poisson Processes Recall, the Poisson random variable counts the number of events occurring in a time (or space) interval where λ (a parameter of the distribution) is the rate of the occurrence of the events per one unit of time (or space). Very often we are interested in the number of events occurring in t units of “time”, “space”, “area”, or “volume”. The model for that case is the Poisson Process. It has the following properties: The number of outcomes occurring in one time interval (or other specified region) is independent of the number that occurs in any other disjoint time interval. Process possessing this property is said to have no memory. The probability that a single outcome will occur during a very short time interval is proportional to the length of the time interval and does not depend on the number of outcomes occurring outside this time interval. The probability that more then one outcome will occur in such a short time interval is negligible. The probability distribution for the random variable that counts the number of events per t units of time is given by… STA286 week 5

The Uniform distribution X has a uniform[0,1] distribution. The pdf of X is given by: In general, if X has a Uniform[a, b] distribution, b > a. The pdf of X is given by: The mean and variance of the Uniform distribution are …. STA286 week 5

The Exponential Distribution A random variable X that counts the waiting time for rare phenomena has Exponential(λ) distribution. The parameter of the distribution λ = average number of occurrences per unit of time (space etc.). The pdf of X is given by: Questions: Is this a valid pdf? What is the cdf of X? Note: The textbook uses different parameterization λ = 1/β. STA286 week 5

Important Facts about Exponential Distribution The Exponential random variable possess and important property called Memoryless property. It is described as follows: The Exponential distribution is often used to describe lifetime of machines or other devices. (Read more on Section 6.7). The mean and Variance of the Exponential distribution are… The Exponential distribution very often describe the time until the occurrence of a Poisson event or the time between Poisson events. STA286 week 5

More on Poisson Processes Model for times of occurrences (“arrivals”) of rare phenomena where λ – average number of arrivals per one unit of time period. X – number of arrivals in t units of time period. How long do I have to wait until the first arrival? Let Y = waiting time for the first arrival (a continuous r.v.) then we have Therefore, which is the CDF of the Exponential distribution. The waiting time for the first occurrence of an event when the number of events follows a Poisson distribution is Exponentially distributed. STA286 week 5

The Gamma distribution A random variable X is said to have a gamma distribution with parameters α > 0 and λ > 0 if and only if the density function of X is where Note: the quantity г(α) is known as the gamma function. It is defined for α > 0 and has the following properties: г(1) = 1 г(α + 1) = α г(α) г(n) = (n – 1)! if n is an integer. STA286 week 5