Discrete and Continuous Random Variables

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Duane Price
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1 Discrete and Continuous Random Variables Friday, January 31, :03 PM Homework 1 will be posted over the weekend; due in a couple of weeks. Important examples of random variables on discrete state spaces 1. Random variables on finite state spaces. Then, up to isomorphism, one can number the states as (or other choices like ) would be a way to represent a state space with a finite number M elements. One then simply defines a probability distribution for such that and. We had an example of this in last lecture for the disease state of an immigrant from a country with endemic TB, with M=3. a. By the definition of some textbooks, the example I gave might not be considered a random variable because some books require that state spaces always be subsets of real or complex Euclidean space. b. An important special case of a probability distribution on finite state spaces is the uniform distribution (aka classical probability models) where all possible values occur with equal probability. That is, if the state space has cardinality, then for all. It's just like M-sided dice in Dungeons and Dragons, whatever that is. 2. Infinite discrete state spaces are, by definition of "discrete", countably infinite, meaning they can be put into a one-to-one correpsondence with the set of integers. Typical examples are: or or or or a subset of one of these. Integer lattice state spaces could represent discretized space in applications in atomic physics, ecology. Stoch14 Page 1

2 A particularly widely used example of a random variable on a countably infinite discrete state space is the Poisson random variable for The Poisson distribution involves one parameter which must be positive and in fact can be shown to be equal to the mean of the Poisson distribution:. So what's so special about this particular probability distribution? It appears in a wide variety of applications because of the Poisson limit theorem (which is an analogue of the central limit theorem which explains why Gaussian/normal random variables appear so frequently). If one is quantifying the total number of occurrences of some incident, and if the incident itself can be represented as a sum of a large number of independent rare incidents, then the quantity of interest can be well modelled by a Poisson distribution. Common application of this idea: The number of incoming "agents" to a "node" over some time interval customers at a store, cars entering an entrance ramp on an expressway, demands on a server, incoming pulses to a neuron Another important probability distribution on the whole numbers (nonnegative integers) is the geometric distribution: for Geometric distribution also requires one parameter to specify it. The canonical way of thinking about the geometric distribution is that one is taking an unbounded sequence of repeated independent trials, with each trial having a success probability of p. The geometric distribution describes how many failures occur before the first success. Note that you'll see geometric distribution defined with variations of this idea, i.e., the number of trials you need until the first success, in which case x is shifted by 1. This distribution appears in discrete-time stochastic processes because it is the only probability distribution on the set of nonnegative integers that has the Stoch14 Page 2

3 memoryless property. (Lack of memory means that the success or failure probability at a given epoch does not depend on the success or failure at previous epochs.) Random variables on continuous state space One cannot take the tactic of simply defining the probability for the random variable to take any given value in the state space, otherwise, one runs into contradictions due to the uncountable infinities involved. The reason one can't define a probability distribution simply in terms of probabilities of singletons is the same reason that in continuum mechanics, one does not try to talk about the mass at a given point. The resolution is to describe the continuously distributed quantity in terms of density (i.e., mass density in continuum solid or fluid mechanics). One can do the same for a broad range of continuously distributed random variables. Definition: An absolutely continuously distributed random variable X is a random variable whose probability distribution can be expressed in terms of a function, called a probability density which has the property that: for all where and In practice, that means one can define an absolutely continuous random variable by specifying any probability density satisfying the above properties. Beware that the probability density is not a probability! So there's no reason that Some important examples (for us) of absolutely continuous random variable models are: 1. Uniform distribution (X is uniformly distributed on the interval [a,b]), Stoch14 Page 3

4 Or equivalently (perhaps more commonly), one takes, 2. Probabilities are obtained by integrating the probability density over the set of interest. The built in random number generator in most operating systems and software distributions is a pseudorandom algorithm for simulating U(0,1). Exponential distribution is another particularly useful absolutely continuous probability distribution. or where is a single parameter needed to define the distribution. One can check that Stoch14 Page 4

5 Now we have a nonuniform probability density, what does it actually mean? To be really precise, one should always talk about the probability for the random variable to fall in nice (Borel) sets like intervals, and for a given interval size, the probability the random variable will fall in that interval is generally larger if the probability density is larger there. And if one wants to give the probability density itself meaning, then one can show using the mean value theorem that, if is continuous at then: The exponential distribution arises frequently in stochastic processes when one is describing the amount of time to wait until some random event occurs (in continuous time) when the waiting time has a memoryless property meaning that the amount of time you have left to wait does not depend on how long you have been waiting. In mathematical terms this means that if is a random variable that is memoryless in this sense, we want: Stoch14 Page 5

6 The exponential distribution is the only absolutely continuous probability distribution that satisfies this memoryless equation, and the geometric distribution is the only discrete probability distribution that satisfies this equation. It's no accident they're both self-similar distributions. And this can be formulated as a rigorous theorem. Of course Gaussian/normal distributions are another very widely used class of absolutely continuous probability distributions, but we won't see much of them in this class. Not every random variable need be discrete or absolutely continuous. There are hybrid random variables that are neither, but can appear in application. And even nastier cases of singular continuous random variables that don't fit in either framework, and do appear in some (but not many) applications like the spectra of random media. Stoch14 Page 6

I will post Homework 1 soon, probably over the weekend, due Friday, September 30.

Random Variables Friday, September 09, 2011 2:02 PM I will post Homework 1 soon, probably over the weekend, due Friday, September 30. No class or office hours next week. Next class is on Tuesday, September