Now let’s go over what a random process is.Ī random process is a sequence of random variables X1, X2, X3,… etc. Real world examples of a purely continuous random variable are not easy to find. The probability distribution of a continuous random variable is called the Probability Density Function (PDF). A close approximation is the temperature of a place at a specific time of the year, measured to some arbitrarily large precision. Real world examples of a purely continuous Xare not easy to find. The probability distribution of a discrete random variable is called the Probability Mass Function (PMF).Ī continuous random variable’s range is the set of real numbers, i.e. A real world example of a discrete X is the number of cars passing through an intersection during some interval of time. The range of a discrete random variable is countably infinite, for e.g. The Probability Mass Function of X (Image by Author) Range( X ): The range of X is the set of real numbers. For example (Heads, Heads) or (Tails, Heads) are two possible outcomes of the coin toss experiment. These outcomes arise when some stochastic experiment is performed (such as tossing a pair of coins). A random variable, usually denoted by X, Y, Z, X1, X2, Z3, etc., is actually a function! And like all well behaved functions, X has a domain and a range.ĭomain( X ): The domain of X is the sample space of random outcomes. The word ‘variable’ in random variable is a misnomer. Now that I have tickled your curiosity, let’s begin our journey into the wonderful world of Poisson processes.īut first, a quick overview of random variables and random processes. Pretty much any event that generates a sequence of whole numbered counts is a candidate for being modeled as a Poisson process. Number of meteors detected per hour during the Perseid meteor shower.Number of electrical pulses generated by a photo-detector that is exposed to a beam of photons, in 1 minute.The number of vehicles passing through some intersection from 8am to 11am on weekdays.Number of failures of ultrasound machines in a hospital over some period of time.The number of hot dogs sold by say, Papaya King, from 12pm to 4pm on Sundays.At a drive-through pharmacy, the number of cars driving up to the drop off window in some interval of time.
Poisson processes can be seen in all walks of life. It turns out such “arrivals” data can be modeled very nicely using a Poisson process.
While creating the above simulation, we have assumed that the average arrival rate is 5 patients per hour. The Y-axis shows the simulated time at which that patient arrived at a hospital’s Emergency Room. A sample Poisson process (Image by Author)