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Realization of a random variable

by , PhD

The value that a random variable X will take is, a priori, unknown. However, after we receive the information that X has taken a certain value x (i.e., $X=x$), the value x is called the realization of X.

The concept extends in the obvious manner also to random vectors and random matrices.

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Note that random variables are usually denoted by an uppercase letter, while their realizations are usually denoted by a lowercase letter.

So, for example, if we denote a random variable by X, its realization is denoted by x.

Remember that random variables are functions defined on the set of all possible outcomes of a probabilistic experiment, which is denoted by Omega and is called sample space.

If omega in Omega is the realized outcome, that is, the outcome that has actually happened, then the value taken by the random variable, which is associated to that outcome, is denoted by[eq1]

Also in this case we say that [eq2] is the realization of X.


Suppose that we perform a probabilistic experiment having two possible outcomes, $\omega _{1}$ and $\omega _{2}$.

The sample space, that is, the set of all possible outcomes, is[eq3]

Now, suppose that we have a random variable X defined as follows:[eq4]

If the realized outcome of the experiment is $\omega _{1}$, then the realization of the random variable X is[eq5]

More details

For more details about the realization of a random variable, you can read the lecture entitled Random variables.

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How to cite

Please cite as:

Taboga, Marco (2021). "Realization of a random variable", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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