This is a beta course, so its structure, chapters, and examples may continue to change.

Random Variables and Distributions

Definition of a Random Variable

The Concept of a Random Variable

Let $\Omega$ be the sample space of a random experiment. If for every sample point $\omega$ in $\Omega$ there is a real number $X(\omega)$ corresponding to it, then $X(\omega)$ is called a random variable, abbreviated as $X$ .

Classification of Random Variables

Classification:

Discrete random variable: a random variable whose values are finitely many or countably infinitely many
Continuous random variable: a random variable whose values are all real numbers on some interval
Mixed random variable: a random variable that is neither discrete nor continuous

Examples of Random Variables

Example 1: rolling a die, with $X$ denoting the number that appears

$X$ is a discrete random variable
The possible values of $X$ are: $1, 2, 3, 4, 5, 6$

Example 2: measuring the length of an object, with $X$ denoting the measurement result

$X$ is a continuous random variable
The possible values of $X$ are: $[0, +\infty)$

Example 3: tossing a coin, with $X$ denoting the number of heads that appear

$X$ is a discrete random variable
The possible values of $X$ are: $0, 1$

Distribution Law of a Discrete Random Variable

Definition of the Distribution Law

Let $X$ be a discrete random variable whose possible values are $x_1, x_2, \dots, x_n, \dots$ . Then: $P(X = x_i) = p_i, \quad i = 1, 2, \dots, n, \dots$

is called the distribution law or probability distribution of $X$ .

Properties of the Distribution Law

Property 1: $p_i \geq 0$ , $i = 1, 2, \dots$

Property 2: $\sum_{i=1}^{\infty} p_i = 1$

Ways of Representing the Distribution Law

Method 1: tabular form

$X$	$x_1$	$x_2$	$\dots$	$x_n$	$\dots$
$P$	$p_1$	$p_2$	$\dots$	$p_n$	$\dots$

Method 2: formula form $P(X = x_i) = p_i, \quad i = 1, 2, \dots$

Examples of Distribution Laws

Example 4: rolling a fair die, with $X$ denoting the number that appears

Distribution law: $P(X = i) = \frac{1}{6}$ , $i = 1, 2, 3, 4, 5, 6$

Example 5: tossing a coin, with $X$ denoting the number of heads that appear

Distribution law: $P(X = 0) = \frac{1}{2}$ , $P(X = 1) = \frac{1}{2}$

Probability Density Function of a Continuous Random Variable

Definition of the Probability Density Function

Let $X$ be a continuous random variable. If there exists a non-negative function $f(x)$ such that for any real numbers $a < b$ : $P(a < X \leq b) = \int_a^b f(x) dx$

then $f(x)$ is called the probability density function of $X$ , or simply the density function.

Properties of the Probability Density Function

Property 1: $f(x) \geq 0$ , $-\infty < x < +\infty$

Property 2: $\int_{-\infty}^{+\infty} f(x) dx = 1$

Property 3: $P(a < X \leq b) = \int_a^b f(x) dx$

Examples of Probability Density Functions

Example 6: a random variable uniformly distributed on the interval $[0, 1]$

Density function: $f(x) = \begin{cases} 1, & 0 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases}$

Example 7: a random variable with the standard normal distribution

Density function: $f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ , $-\infty < x < +\infty$

Distribution Function

Definition of the Distribution Function

Let $X$ be a random variable. The function: $F(x) = P(X \leq x), \quad -\infty < x < +\infty$

is called the distribution function of $X$ .

Properties of the Distribution Function

Property 1: $0 \leq F(x) \leq 1$ , $-\infty < x < +\infty$

Property 2: $F(x)$ is a non-decreasing function

Property 3: $\lim_{x \to -\infty} F(x) = 0$ , $\lim_{x \to +\infty} F(x) = 1$

Property 4: $F(x)$ is a right-continuous function

Relation Between the Distribution Function and the Distribution Law

Discrete random variable: $F(x) = \sum_{x_i \leq x} P(X = x_i)$

Relation Between the Distribution Function and the Density Function

Continuous random variable: $F(x) = \int_{-\infty}^x f(t) dt$

Derivative relation: $f(x) = F'(x)$

Examples of Distribution Functions

Example 8: the distribution function of a die roll

F(x) = \begin{cases} 0, & x < 1 \\ \frac{1}{6}, & 1 \leq x < 2 \\ \frac{2}{6}, & 2 \leq x < 3 \\ \frac{3}{6}, & 3 \leq x < 4 \\ \frac{4}{6}, & 4 \leq x < 5 \\ \frac{5}{6}, & 5 \leq x < 6 \\ 1, & x \geq 6 \end{cases}

Example 9: the distribution function of the uniform distribution $[0, 1]$

F(x) = \begin{cases} 0, & x < 0 \\ x, & 0 \leq x < 1 \\ 1, & x \geq 1 \end{cases}

Common Discrete Distributions

Two-Point Distribution (Bernoulli Distribution)

If the random variable $X$ takes only the two values $0$ and $1$ , with $P(X = 1) = p$ and $P(X = 0) = 1-p$ , then $X$ is said to follow the two-point distribution with parameter $p$ , denoted by $X \sim B(1, p)$ .

Distribution law: $P(X = k) = p^k(1-p)^{1-k}, \quad k = 0, 1$

Binomial Distribution

If the random variable $X$ denotes the number of occurrences of event $A$ in $n$ independent repeated trials, and $P(A) = p$ in every trial, then $X$ is said to follow the binomial distribution with parameters $n, p$ , denoted by $X \sim B(n, p)$ .

Distribution law: $P(X = k) = C_n^k p^k(1-p)^{n-k}, \quad k = 0, 1, 2, \dots, n$

Poisson Distribution

If the distribution law of the random variable $X$ is: $P(X = k) = \frac{\lambda^k}{k!} e^{-\lambda}, \quad k = 0, 1, 2, \dots$

then $X$ is said to follow the Poisson distribution with parameter $\lambda$ , denoted by $X \sim P(\lambda)$ .

Common Continuous Distributions

Uniform Distribution

If the probability density function of the random variable $X$ is:

f(x) = \begin{cases} \frac{1}{b-a}, & a \leq x \leq b \\ 0, & \text{otherwise} \end{cases}

then $X$ is said to follow the uniform distribution on the interval $[a, b]$ , denoted by $X \sim U(a, b)$ .

Exponential Distribution

If the probability density function of the random variable $X$ is:

f(x) = \begin{cases} \lambda e^{-\lambda x}, & x \geq 0 \\ 0, & x < 0 \end{cases}

then $X$ is said to follow the exponential distribution with parameter $\lambda$ , denoted by $X \sim E(\lambda)$ .

Normal Distribution

If the probability density function of the random variable $X$ is: $f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}, \quad -\infty < x < +\infty$

then $X$ is said to follow the normal distribution with parameters $\mu, \sigma^2$ , denoted by $X \sim N(\mu, \sigma^2)$ .

Exercises

Exercise 1

Given the distribution law $P(X=1)=0.2, P(X=2)=0.5, P(X=3)=0.3$ of the random variable $X$ , find the distribution function of $X$ .

Reference Answer

distribution functiondefinition and computationcontinuous random variable

Approach: Compute from the definition of the distribution function.

Steps:

$F(x) = P(X \leq x) = \sum_{x_i \leq x} P(X = x_i)$
When $x < 1$ , $F(x) = 0$
When $1 \leq x < 2$ , $F(x) = P(X = 1) = 0.2$
When $2 \leq x < 3$ , $F(x) = P(X = 1) + P(X = 2) = 0.2 + 0.5 = 0.7$
When $x \geq 3$ , $F(x) = P(X = 1) + P(X = 2) + P(X = 3) = 0.2 + 0.5 + 0.3 = 1$

Answer:

F(x) = \begin{cases} 0, & x < 1 \\ 0.2, & 1 \leq x < 2 \\ 0.7, & 2 \leq x < 3 \\ 1, & x \geq 3 \end{cases}

Exercise 2

Given that the distribution function of the random variable $X$ is $F(x)=\begin{cases} 0, x<0 \\ x, 0\leq x<1 \\ 1, x\geq1 \end{cases}$ , find the probability density function of $X$ .

Reference Answer

probability density functiondistribution functioncontinuous random variable

Approach: Use the fact that the density function is the derivative of the distribution function.

Steps:

$f(x) = F'(x)$
When $x < 0$ , $F(x) = 0$ , so $f(x) = 0$
When $0 \leq x < 1$ , $F(x) = x$ , so $f(x) = 1$
When $x \geq 1$ , $F(x) = 1$ , so $f(x) = 0$

Answer:

f(x) = \begin{cases} 1, & 0 \leq x < 1 \\ 0, & \text{otherwise} \end{cases}

Exercise 3

Roll two dice, with $X$ denoting the sum of the numbers on the two dice. Find the distribution law of $X$ .

Reference Answer

definition of probabilityclassical probability modelrandom experiment

Approach: List all possible outcomes and compute their probabilities.

Steps:

The possible values of the sum of the two dice are: $2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12$
Compute the probability of each case:
- $P(X = 2) = \frac{1}{36}$ (only the case (1,1))
- $P(X = 3) = \frac{2}{36}$ (the cases (1,2), (2,1))
- $P(X = 4) = \frac{3}{36}$ (the cases (1,3), (2,2), (3,1))
- and so on…

Answer:

$X$	2	3	4	5	6	7	8	9	10	11	12
$P$	$\frac{1}{36}$	$\frac{2}{36}$	$\frac{3}{36}$	$\frac{4}{36}$	$\frac{5}{36}$	$\frac{6}{36}$	$\frac{5}{36}$	$\frac{4}{36}$	$\frac{3}{36}$	$\frac{2}{36}$	$\frac{1}{36}$

Exercise 4

Given that the random variable $X$ follows an exponential distribution with parameter $\lambda$ , find the distribution function of $X$ .

Reference Answer

probability density functiondistribution functioncontinuous random variable

Approach: Use the relation between the distribution function and the density function.

Steps:

The density function of the exponential distribution: $f(x) = \begin{cases} \lambda e^{-\lambda x}, & x \geq 0 \\ 0, & x < 0 \end{cases}$
The distribution function: $F(x) = \int_{-\infty}^x f(t) dt$
When $x < 0$ , $F(x) = 0$
When $x \geq 0$ , $F(x) = \int_0^x \lambda e^{-\lambda t} dt = 1 - e^{-\lambda x}$

Answer:

F(x) = \begin{cases} 0, & x < 0 \\ 1 - e^{-\lambda x}, & x \geq 0 \end{cases}

Exercise 5

Prove that for any random variable $X$ , its distribution function $F(x)$ satisfies $0 \leq F(x) \leq 1$ .

Reference Answer

distribution functionpropertiesproof

Approach: Use the definition of the distribution function and the properties of probability.

Steps:

$F(x) = P(X \leq x)$
By the non-negativity of probability, $P(X \leq x) \geq 0$ , so $F(x) \geq 0$
By the normalization of probability, $P(X \leq x) \leq 1$ , so $F(x) \leq 1$
Therefore $0 \leq F(x) \leq 1$

Answer: the proof is complete

Numerical Characteristics of Random Variables

Basic Concepts of Mathematical Expectation

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Random Variables and Distributions

Definition of a Random Variable

The Concept of a Random Variable

Classification of Random Variables

Examples of Random Variables

Distribution Law of a Discrete Random Variable

Definition of the Distribution Law

Properties of the Distribution Law

Ways of Representing the Distribution Law

Examples of Distribution Laws

Probability Density Function of a Continuous Random Variable

Definition of the Probability Density Function

Properties of the Probability Density Function

Examples of Probability Density Functions

Distribution Function

Definition of the Distribution Function

Properties of the Distribution Function

Relation Between the Distribution Function and the Distribution Law

Relation Between the Distribution Function and the Density Function

Examples of Distribution Functions

Common Discrete Distributions

Two-Point Distribution (Bernoulli Distribution)

Binomial Distribution

Poisson Distribution

Common Continuous Distributions

Uniform Distribution

Exponential Distribution

Normal Distribution

Exercises

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5