Random Variables and Distributions

This chapter continues developing the theory of probability by defining random variables that are functions defined on the sample space of a probability space. Most stochastic models are expressed in terms of random variables: the number of customers in a queue, the fraction of time a processor is busy, and the amount of time there are fewer than k customers in the network are all examples of random variables. It is typically more convenient to work with random variables than with the underlying basic events of a sample space. Unlike a variable in algebra, which represents a deterministic value, a random variable represents the outcome of a random experiment and can thus only be characterized by its probabilistic outcome. To characterize these possibilities, we specify a distribution function that expresses the probability that a random variable has a value within a certain range. For example, if we let X k be the number of heads obtained in k tosses of a fair coin, then the value of X k is random and can range from 0 to k. The probability that there are less than or equal to ℓ heads, denoted by P [X k ≤ ℓ], can be calculated by enumerating all possible outcomes and is given by 4.1 % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacU % facaWGybWaaSbaaSqaaiaadUgaaeqaaOGaeyizImQaeS4eHWMaaiyx % aiabg2da9iaaikdadaahaaWcbeqaaiabgkHiTiaadUgaaaGcdaaeWb % qaamaabmaaeaqabeaacaWGRbaabaGaamyAaaaacaGLOaGaayzkaaaa % leaacaWGPbGaeyypa0JaaGimaaqaaiabloriSbqdcqGHris5aaaa!4AD8! $$P[{X_k} \leqslant \ell ] = {2^{ - k}}\sum\limits_{i = 0}^\ell {\left( \begin{gathered} k \hfill \\ i \hfill \\ \end{gathered} \right)}$$ (notice that this is the enigmatic summation of (3.36)). The set of values of P [X k ≤ ℓ] for all ℓ is called the distribution function of the random variable X k . In this chapter we will show that a distribution function can be used in place of a probability space when specifying the outcome of a random experiment. This is the basis for most probabilistic models that specify the distributions of random variables of interest but do not specify the underlying probability space. In fact, it is very rare in models to find a specification of a probability space since this is usually understood from the context of the model.