**Definition of Random Variable**

When numbers are associated with the outcome of a chance experiment, the result is said to be a *random variable*. Thus a random variable is a function with simple space *S* = {*s*_{1}, *s*_{2},…,s* _{N}*} as the domain, and a set of real numbers

*X*= {X

_{(S1)},

*X*

_{(S2)},…,

*X*

_{(SN)}} as the range.

Some possible random variables we might define are the following :

1. **The number of successes**. The random variable *X = *number of successes is the quantity *k* described by the binomial distribution, *B*_{5}(*k,p*)

2. **The trial on which the kth success occurs. **The random variable

*X*= trial for the

*k*th success is the quantity

*n*described by the Pascal distribution,

*P*, except that we must assign the value to

_{k}(n,p)*X*in the event that no

*k*th success occurs.

3. **The number of failures subtracted from the number of successes. **This distribution we have not investigated, but it is easily treated.

4. **Binary numbers.** ** **Another possible random variable is *X* = numerical value of the outcome considering *A* as 1 and as 0 and interpreting the number in base 2.

** **

*X*** and x ?**

We must distinguish between *X(s)*, the random variable, and *x*, an independent, continuous mathematical variable ranging over the real line. For example, we set *x = *0 and proceed as follows. We perform the experiment of 5 binomial trials. An outcome occurs. That outcome has 0, 1, 2, 3, 4, or 5 successes. The corresponding value of *X(s)* is the number of successes, a random variable. If *X(s) *= 0, then the event {*X(s) = x*} occurred since *x* was set to 0. If 1, 2, 3, 4, or 5 successes occurred, then the event *X(s) = x* did not occur. For *x = *1.3, the event {*X = x*} never occurs.

**Benefit of introducing random variables**

1. Numbers come up naturally in most practical and many theoretical problems. Usually we’re interested in some sort of numerical information.

2. Random variables make all experiments alike. The concept introduces a common basis for describing random experiments.

3. Random variables allow us some distance from the chance experiment. In many situation the random variable can be easily identified and measured, whereas the chance experiment may be quite abstract and difficult to define.

I’m getting dizzy with this kind of lesson…what a difficult statistics…! But please understand this…

Taken from the *Modelling Random System* by J.R. Cogdell