Discrete probability distributions are essential tools for modeling random events in stochastic processes. They describe the likelihood of different outcomes for variables that take on distinct, countable values. Understanding these distributions is crucial for analyzing and predicting behavior in various systems.

This section covers key discrete distributions like Bernoulli, binomial, geometric, and Poisson. It explores probability mass functions, cumulative distribution functions, expected values, and variance. These concepts form the foundation for analyzing discrete random variables in stochastic processes.

Types of discrete distributions

• Discrete probability distributions are a fundamental concept in the study of stochastic processes, which model real-world scenarios involving random variables that take on discrete values
• Understanding the various types of discrete distributions is essential for analyzing and predicting the behavior of stochastic systems, such as queueing systems, inventory management, and reliability engineering
• Key discrete distributions covered in this section include Bernoulli, binomial, geometric, negative binomial, Poisson, and hypergeometric distributions

Probability mass functions

• Probability mass functions (PMFs) are a crucial tool for describing and quantifying the probability distribution of discrete random variables in stochastic processes

Definition of PMF

• A probability mass function (PMF) is a function that assigns probabilities to each possible value of a discrete random variable
• Denoted as $P(X = x)$, where $X$ is the random variable and $x$ is a specific value that $X$ can take
• The PMF gives the probability that the random variable $X$ takes on the value $x$

Properties of valid PMFs

• The PMF must be non-negative for all possible values of the random variable, i.e., $P(X = x) \geq 0$ for all $x$
• The sum of probabilities for all possible values of the random variable must equal 1, i.e., $\sum_{x} P(X = x) = 1$
• These properties ensure that the PMF is a valid probability distribution and can be used to model stochastic processes

Cumulative distribution functions

• Cumulative distribution functions (CDFs) provide an alternative way to describe the probability distribution of discrete random variables and are particularly useful for calculating probabilities of events involving inequalities

Definition of CDF

• The cumulative distribution function (CDF) of a discrete random variable $X$ is denoted as $F(x)$ and is defined as the probability that $X$ takes on a value less than or equal to $x$
• Mathematically, $F(x) = P(X \leq x) = \sum_{t \leq x} P(X = t)$, where $t$ represents all possible values of $X$ less than or equal to $x$
• The CDF is a non-decreasing function, meaning that $F(a) \leq F(b)$ for all $a \leq b$

Relationship between PMF and CDF

• The PMF and CDF are related, and one can be derived from the other
• For a discrete random variable $X$, the PMF can be obtained from the CDF by taking the difference between consecutive CDF values: $P(X = x) = F(x) - F(x-1)$
• Conversely, the CDF can be obtained from the PMF by summing the PMF values up to and including the desired value: $F(x) = \sum_{t \leq x} P(X = t)$

Expected value

• The expected value, also known as the mean or expectation, is a key concept in stochastic processes that summarizes the average behavior of a discrete random variable

Definition of expected value

• The expected value of a discrete random variable $X$ is denoted as $E[X]$ and is defined as the weighted average of all possible values of $X$, where the weights are the corresponding probabilities
• Mathematically, $E[X] = \sum_{x} x \cdot P(X = x)$, where $x$ represents all possible values of $X$
• The expected value provides a measure of the central tendency of the random variable and is useful for making predictions and comparing different distributions

Linearity of expectation

• The linearity of expectation is an important property that simplifies the calculation of expected values for sums of random variables
• For any two discrete random variables $X$ and $Y$, the expected value of their sum is equal to the sum of their individual expected values: $E[X + Y] = E[X] + E[Y]$
• This property holds regardless of whether $X$ and $Y$ are independent or dependent, making it a powerful tool for analyzing stochastic processes involving multiple random variables

Variance and standard deviation

• Variance and standard deviation are measures of the dispersion or spread of a discrete random variable around its expected value, providing insight into the variability of the stochastic process

Definition of variance

• The variance of a discrete random variable $X$ is denoted as $Var(X)$ or $\sigma^2$ and is defined as the expected value of the squared deviation from the mean
• Mathematically, $Var(X) = E[(X - E[X])^2] = \sum_{x} (x - E[X])^2 \cdot P(X = x)$, where $x$ represents all possible values of $X$
• A higher variance indicates greater dispersion of the random variable around its expected value

Properties of variance

• Variance has several important properties that facilitate its calculation and interpretation
• For any constant $a$ and discrete random variable $X$, $Var(aX) = a^2 Var(X)$
• For any two independent discrete random variables $X$ and $Y$, $Var(X + Y) = Var(X) + Var(Y)$
• These properties allow for the decomposition and simplification of variance calculations in stochastic processes

Standard deviation vs variance

• The standard deviation, denoted as $\sigma$, is the square root of the variance: $\sigma = \sqrt{Var(X)}$
• While variance is measured in squared units, standard deviation is measured in the same units as the random variable, making it more interpretable
• Standard deviation is often preferred when comparing the dispersion of different distributions or communicating results to non-technical audiences

Moment generating functions

• Moment generating functions (MGFs) are a powerful tool for characterizing and analyzing discrete probability distributions, providing an alternative approach to working with PMFs and CDFs

Definition of MGF

• The moment generating function of a discrete random variable $X$ is denoted as $M_X(t)$ and is defined as the expected value of $e^{tX}$
• Mathematically, $M_X(t) = E[e^{tX}] = \sum_{x} e^{tx} \cdot P(X = x)$, where $x$ represents all possible values of $X$ and $t$ is a real number
• The MGF uniquely determines the probability distribution of $X$ and can be used to derive various properties of the distribution

Properties and applications of MGFs

• MGFs have several useful properties that facilitate the analysis of discrete probability distributions
• The $n$-th derivative of the MGF evaluated at $t = 0$ gives the $n$-th moment of the distribution: $M_X^{(n)}(0) = E[X^n]$
• For independent random variables $X$ and $Y$, the MGF of their sum is the product of their individual MGFs: $M_{X+Y}(t) = M_X(t) \cdot M_Y(t)$
• MGFs can be used to derive the PMF, CDF, expected value, variance, and other properties of a distribution, making them a versatile tool in stochastic processes

Common discrete distributions

• Several discrete probability distributions frequently arise in the study of stochastic processes, each with its own unique properties and applications

Bernoulli and binomial distributions

• The Bernoulli distribution models a single trial with two possible outcomes (success or failure), with success probability $p$
• The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, with parameters $n$ (number of trials) and $p$ (success probability)
• Binomial random variables are commonly used to model systems with repeated trials, such as quality control and survey sampling

Geometric and negative binomial distributions

• The geometric distribution models the number of trials until the first success in a sequence of independent Bernoulli trials, with success probability $p$
• The negative binomial distribution generalizes the geometric distribution, modeling the number of trials until the $r$-th success in a sequence of independent Bernoulli trials, with parameters $r$ and $p$
• These distributions are useful for modeling waiting times and the number of failures before a specified number of successes in stochastic processes

Poisson distribution

• The Poisson distribution models the number of events occurring in a fixed interval of time or space, given an average rate of occurrence $\lambda$
• Poisson random variables are used to model systems with rare events occurring independently and at a constant rate, such as customer arrivals in a queueing system or defects in a manufacturing process

Hypergeometric distribution

• The hypergeometric distribution models the number of successes in a fixed number of draws from a finite population without replacement, with parameters $N$ (population size), $K$ (number of successes in the population), and $n$ (number of draws)
• This distribution is applicable in scenarios where sampling is performed without replacement, such as quality control and audit sampling

Joint distributions

• Joint distributions describe the simultaneous behavior of two or more discrete random variables, enabling the analysis of their relationships and dependencies in stochastic processes

Joint probability mass functions

• The joint probability mass function (joint PMF) of two discrete random variables $X$ and $Y$ is denoted as $P(X = x, Y = y)$ and gives the probability that $X$ takes on the value $x$ and $Y$ takes on the value $y$ simultaneously
• The joint PMF must satisfy the properties of a valid PMF, such as non-negativity and summing to 1 over all possible pairs of values $(x, y)$

Marginal and conditional distributions

• Marginal distributions describe the behavior of a single random variable in a joint distribution, obtained by summing the joint PMF over all possible values of the other variable(s)
• For discrete random variables $X$ and $Y$, the marginal PMF of $X$ is given by $P(X = x) = \sum_{y} P(X = x, Y = y)$
• Conditional distributions describe the behavior of one random variable given the value of another, obtained by dividing the joint PMF by the relevant marginal PMF
• The conditional PMF of $Y$ given $X = x$ is given by $P(Y = y | X = x) = \frac{P(X = x, Y = y)}{P(X = x)}$

Independent vs dependent random variables

• Two discrete random variables $X$ and $Y$ are independent if their joint PMF can be factored into the product of their marginal PMFs: $P(X = x, Y = y) = P(X = x) \cdot P(Y = y)$ for all $x$ and $y$
• If $X$ and $Y$ are independent, their conditional PMFs are equal to their marginal PMFs: $P(Y = y | X = x) = P(Y = y)$ and $P(X = x | Y = y) = P(X = x)$
• Dependent random variables have joint PMFs that cannot be factored into the product of their marginal PMFs, indicating a relationship between the variables that must be accounted for in stochastic process modeling

Sums of discrete random variables

• Calculating the distribution of the sum of discrete random variables is a common task in stochastic processes, particularly when analyzing systems with multiple components or stages

Convolution formula for PMFs

• The convolution formula is a method for determining the PMF of the sum of two independent discrete random variables $X$ and $Y$
• Denoting the sum as $Z = X + Y$, the PMF of $Z$ is given by $P(Z = z) = \sum_{x} P(X = x) \cdot P(Y = z - x)$, where the sum is taken over all possible values of $x$
• The convolution formula can be extended to sums of more than two random variables by repeatedly applying the formula to pairs of variables

Distribution of sum of independent variables

• When the discrete random variables being summed are independent and identically distributed (i.i.d.), the distribution of their sum can often be determined using known properties of the individual distributions
• For example, the sum of $n$ i.i.d. Bernoulli random variables with success probability $p$ follows a binomial distribution with parameters $n$ and $p$
• Similarly, the sum of $n$ i.i.d. Poisson random variables with rate $\lambda$ follows a Poisson distribution with rate $n\lambda$
• Exploiting these properties can simplify the analysis of stochastic processes involving sums of i.i.d. random variables

Transformations of discrete random variables

• Transforming discrete random variables involves creating new random variables as functions of existing ones, which is useful for modeling stochastic processes with complex relationships between variables

PMF and CDF under transformations

• To determine the PMF or CDF of a transformed discrete random variable $Y = g(X)$, where $g$ is a function and $X$ is a discrete random variable with known PMF, we need to consider the preimage of each possible value of $Y$ under $g$
• The PMF of $Y$ is given by $P(Y = y) = \sum_{x: g(x) = y} P(X = x)$, where the sum is taken over all values of $x$ such that $g(x) = y$
• The CDF of $Y$ can be obtained by summing the PMF of $Y$ up to the desired value, as discussed earlier

Functions of multiple discrete variables

• When transforming multiple discrete random variables, the joint PMF of the transformed variables can be determined using a similar approach
• For a transformation $(U, V) = (g_1(X, Y), g_2(X, Y))$, where $X$ and $Y$ are discrete random variables with known joint PMF, the joint PMF of $(U, V)$ is given by $P(U = u, V = v) = \sum_{(x, y): g_1(x, y) = u, g_2(x, y) = v} P(X = x, Y = y)$
• The marginal and conditional PMFs of the transformed variables can be obtained from the joint PMF using the methods discussed earlier

Applications and examples

• Discrete probability distributions find numerous applications in modeling and analyzing real-world stochastic processes across various domains

Modeling real-world scenarios

• Queueing systems: Discrete distributions can model customer arrivals (Poisson), service times (geometric or negative binomial), and system performance measures (binomial)
• Inventory management: Demand for products can be modeled using discrete distributions (Poisson or binomial), enabling the optimization of inventory levels and reorder points
• Reliability engineering: The number of defects or failures in a system can be modeled using discrete distributions (binomial or Poisson), facilitating the assessment of system reliability and maintenance strategies

Solving problems using discrete distributions

• Quality control: Discrete distributions can be used to determine the probability of accepting or rejecting a lot based on the number of defective items found in a sample (hypergeometric)
• Risk assessment: The probability and severity of adverse events can be modeled using discrete distributions (Poisson or binomial), enabling the quantification and management of risk in various contexts
• Network analysis: Discrete distributions can model the degree distribution of nodes in a network (Poisson or power-law), providing insights into network structure and resilience
• By leveraging the properties and relationships of discrete probability distributions, stochastic processes can be effectively modeled, analyzed, and optimized to support decision-making in a wide range of applications.