Random Variables
Related Concepts
Random Variable
Given a probability space \((\Omega, \mathcal{F}, P)\), a function \(X : \Omega \to \mathbb{R}\) defined on the sample space \(\Omega\) is called a random variable if for any \(t \in \mathbb{R}\), we have
Indicator Function
For an event \(A\) on the sample space \(\Omega\), define the random variable
\(I_A\) is called the indicator function of event \(A\).
Distribution Function
For a random variable \(X\), the function
is called the distribution function of \(X\), denoted as \(X \sim F(x)\).
The distribution function has the following properties:
- Right-continuity: \(F(x) = F(x + 0)\).
- Monotonicity: Non-decreasing on \(\mathbb{R}\).
- \(F(-\infty) = 0\), \(F(+\infty) = 1\).
It can also be proved that any function satisfying the above requirements is the distribution function of some random variable. Therefore, there is a one-to-one correspondence between distribution functions and random variables.
Classification of Random Variables
Random variables are classified into discrete and continuous types based on whether their range (since a random variable is a function) is countable.
Discrete Random Variable
Let \(X\) be a discrete random variable with all possible values \(x_1, x_2, \cdots\). Then \(X\) can be described by a series of equalities of the form \(P\{ X = x_i \} = p_i\). This is the probability mass function (or distribution series) learned in high school textbooks.
Continuous Random Variable
For a continuous random variable \(X\), examining \(P\{ X = x \}\) is often meaningless (because this probability is likely \(0\)).
Why is the probability 'likely' \(0\)?
Consider a random variable \(X\) that takes the value \(0\) with probability \(\frac{1}{2}\), and follows a uniform distribution on the open interval \((0, 1)\) with probability \(\frac{1}{2}\). Clearly, \(X\) satisfies the definition of a continuous random variable.
For any real number \(r \in (0, 1)\), it is not difficult to see that \(P\{ X = r \} = 0\), but at the same time \(P\{ X = 0 \} = \frac{1}{2}\).
On the other hand, let \(X \sim F(x)\). Then
A natural idea is to use the limit \(\lim\limits_{\Delta x \to 0^+} \frac{F(l + \Delta x) - F(l)}{\Delta x}\) to describe the likelihood of \(X\) taking the value \(l\).
This expression is the familiar derivative, so the problem reduces to finding a non-negative function \(f(x)\) such that
If such an \(f(x)\) exists, it is called the density function of \(X\).
Independence of Random Variables
We previously discussed the independence of random events. Since random variables are closely related to random events, we can similarly define the independence of random variables.
Definition
Random variables \(X\) and \(Y\) are said to be independent if for any \(x, y \in \mathbb{R}\), we have
Note
Some students might notice that high school textbooks define the independence of random variables using probabilities of the form \(P(X = \alpha)\). However, since continuous random variables usually have probability \(0\) of taking a specific value, it is wiser to define independence using distribution functions in more general cases.
Properties
If random variables \(X\) and \(Y\) are independent, then for any functions \(f, g\), the random variables \(f(X)\) and \(g(Y)\) are independent.
Caution
Sometimes we study the distribution of a function \(f(X, Y)\) (e.g., \(XY^2\)) of independent random variables \(X\) and \(Y\).
Although \(X\) and \(Y\) are independent, one should not assume that for a particular value \(y\) of \(Y\), \(f(X, y)\) and \(f(X, Y)\) follow the same distribution.
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