Someone ask me to explain the different between random variables and random process. Here are my explanation.
Say we are a wholesaler of apples in a city. One day a worker moves down a bucket of apples from a truck. By looking at the apples in this bucket, we can measure the expected weight and variation of apples in this bucket. Let denote the expected weight as , in which it is a random variable that possesses finite mean and variation.
Then, the worker moves down another bucket of apples. We not sure if this bucket of apples possess the same statistical properties as the previous bucket. By intuition, we know these apples in the second bucket possesses a finite expected weight as well (or you will see a HUGE apple!!). Next, the worker continue to move down a lot of buckets of apples from the truck (big truck!). For convenience, we denote the expected weight as where denotes the sequence where the worker moves down the buckets.
The manager of the wholesaler is not interested at the weight distribution of apples in any particular bucket but among certain groups of buckets. For example, he wants to know if the first bucket has worm, what is the chance that the nth bucket has worm as well. This question the inter-relation among the buckets. So, we have no choice but to extend the notion “a single bucket” into “groups of buckets”. In mathematics, we write this as and call this the random process (or stochastic process) of X. In such a way, we can describe the correlation (or other statistical properties) between the first bucket and the second bucket or any bucket in the later sequence to the manager.
As for the summary, in random process (stochastic process) we are dealing with a group of “homogeneous” random variables in the sense that they are of the same function (e.g. apples) but they may possess different distribution function.