Random Variables

Let's begin with an example to see what is a random variable.

Examples of Variables (Exact and Random)

Example of Exact Variable

Suppose that you live in Gotham city and you are in a coffee shop. You open your smartphone and share your location with your girlfriend or boyfriend to find you. Your location at that specific moment is a variable and it has an exact value that can be determined by GPS.

Example of Random Variable

Now imagine another experiment. Your girlfriend or boyfriend is looking for you in Gotham city. She or he does not have a cell phone and could not know your exact location. She or he starts looking for you and she or he knows that you are in one of these 4 places: your house, your office, the coffee shop near your house, Batman's cave! The variable, in this case, is your location from your girlfriend or boyfriend's point of view. This variable could have 4 values and it is not determined before performing the experiment (Your GF or BF looking for you is the experiment). This kind of variable is called Random Variable.

Exact and Random Variables Definition

Based on this example we can define a random variable as follows. First, let's define variables. A variable is a quantity that can have a value and the value can be determined by conducting an experiment. An exact variable is a variable that can have only one possible value and after the experiment, we can find that value. The point is that repetitions of the same experiment will result in the same value for the exact variable. A random variable is a variable that can have multiple values and after the experiment, we get one of those values. The point is that repetitions of the same experiment necessarily will NOT result in the same value for the random variable.

Sources of Randomness

Random variables exist because of the uncertainties in the world. There could be many sources for uncertainties such as:

• Intrinsic uncertainty in nature (uncertainty in quantum mechanics)
• Our limited knowledge about the phenomena's underlying mechanism
• Errors in measurement devices and experiment methods

Random Variables Notations

A random variable in probability is usually denoted by \(X\). The notation \(P(X=x)\) means the probability of random variable \(X\) to have value \(x\) .