2.1 Random variables and probability functions

2.1.1 Random variables

Definition:

  1. A random variable (r.v.) \(X\) is a real-valued (measurable) function which maps the sample space \(\Omega\) to the real numbers \(\mathbb{R}\), i.e. \(X:\Omega \rightarrow \mathbb{R}\). For any sample point \(\omega\in\Omega\), we have \(X(\omega)=x, x\in \mathbb{R}\).

  2. Define the range of a random variable \(X\) is \(R_X = \{ x|x \in X(\omega), \forall \omega\in\Omega \}\)

Remark: The range of a random variable \(X\) is also called support, denoted by \(supp(X)\).

Here we give the definition of random variable in a simple way. For other definition, see Chapter 16.

We can divide the random variables into the following two types:

  1. If \(R_X\) is finite or countable, then it’s a discrete random variable.
  2. If \(R_X\) is uncountable, then it’s a continuous random variable.

Example:

  1. Tossing two fair coins. Define a random variable \(X=\) the number of heads. Then \(\Omega=\{HH,HT,TH,TT\}, R_X=\{0,1,2\}\).

  2. \(X=\) the number of car accident in a day. \(\Omega=\mathbb{N}=R_X\)

  3. \(X=\) the height (cm) of a person. \(\Omega=\mathbb{R}=R_X\) (actually some values are impossible).

2.1.2 Discrete Probability Function

Definition:

Suppose \(X\) is a discrete random variable with range \(R_X\).

  1. Define the function \[ p_X(x) = \begin{cases} P(X=x), & \forall x \in R_X \\ 0, & \forall x \notin R_X \\ \end{cases} \] Then \(p_X(x)\) is called probability mass function (PMF) of random variable \(X\).

  2. Define \(\displaystyle F_X(a) = P(X \leq a) = \sum_{x \leq a} p_X(x), \quad x \in \mathbb{R}\), then \(F_X(x)\) is called the cumulative distribution function (CDF) of random variable \(X\).


Proposition:

  1. \(0 \leq p_X(x) \leq 1, \forall x \in R_X\)
  2. \(\displaystyle \sum_{x \in R_X} p_X(x) = 1\)
  3. \(0 \leq F_X(x) \leq 1, \forall x \in \mathbb{R}\)
  4. \(\displaystyle \lim_{x \to -\infty} F_X(x) = 0, \quad \lim_{x \to \infty} F_X(x) = 1\)
  5. \(\displaystyle \lim_{x \to c^+} F_X(x) = F_X(c^+) = F_X(c), \quad \forall c \in \mathbb{R}\) (right continuous)
  6. If \(a<b\), then \(F_X(a) \leq F_X(b)\) (non-decreasing)
  7. \(F_X(x)\) is a step function
  8. \(p_X(x) = F_X(x)-F_X(x^-)\)
  9. If \(a<b\), then \(\displaystyle P(a < x \leq b) = F_X(b) - F_X(a) = \sum_{a < x \leq b} p_X(x)\)

2.1.3 Continuous Probability Function

Now we disscuss the continuous case

Definition:

Suppose \(X\) is a continuous random variable with range \(R_X\). Then

  1. \(F_X(a) = P(X \leq a), \quad x \in \mathbb{R}\) is called the cumulative distribution function (CDF) of random variable \(X\).

  2. Define \[ f_X(x) = \begin{cases} F'_X(x) = \frac{dF_X(x)}{dx}, & \mbox{if derivative exist at} x \\ 0, & \mbox{otherwise} \\ \end{cases} \] Then \(f_X(x)\) is called probability density function (PDF) of random variable \(X\).


Proposition:

  1. \(f_X(x) \geq 0, \quad \forall x \in R_X\)
  2. \(\int_{x \in R_X} f_X(x) \, dx = 1\)
  3. \(F_X(a) = \int_{-\infty}^a f_X(x) \, dx\)
  4. \(P(X=a) = 0\)
  5. If \(a<b\), then \(P(a<x<b) = F_X(b) - F_X(a) = \int_a^b f_X(x) \, dx\)
  6. \(0 \leq F_X(x) \leq 1, \forall x \in \mathbb{R}\)
  7. \(\displaystyle \lim_{x \to -\infty} F_X(x) = 0, \quad \lim_{x \to \infty} F_X(x) = 1\)
  8. \(\displaystyle \lim_{x \to c^-} F_X(x) = F_X(c^-) = F_X(c) = F_X(c^+) = \lim_{x \to c^+} F_X(x), \quad \forall c \in \mathbb{R}\) (continuous)
  9. If \(a<b\), then \(F_X(a) \leq F_X(b)\) (non-decreasing)

2.1.4 * Mixed Type Probability Function

A part of random variable is discrete, while another part is continuous

Proposition:

Suppose \(X\) is a mixed type random variable. Then \[F_X(x) = aF_d(x) + (1-a)F_c(x), \quad 0<a<1,\] where \(F_d(x)\) is a discrete random variable, \(F_c(x)\) is a continuous random variable.

Example: \[ F_X(x) = \begin{cases} 0, & x<0 \\ x^2+0.1, & 0 \leq x < 0.5 \\ x, & 0.5 \leq x < 1 \\ 1, & x \geq 1 \end{cases} \]

TBD (write into decomposition and find prob. func.)