The Central Idea
A deterministic sequence assigns one fixed value to each index. A random sequence assigns a random variable to each index.
We commonly write a random sequence as
\[ \boxed{X_0,X_1,X_2,\ldots} \]or more compactly as
\[ \boxed{\{X_n\}_{n\ge 0}}. \]The index \(n\) tells us when or where we are in the sequence. The random variable \(X_n\) describes the uncertain value observed at that index.
Review: An Ordinary Sequence
An ordinary numerical sequence may be written
\[ 2,4,6,8,\ldots \]with rule
\[ a_n=2n \]when indexing begins at \(n=1\).
Once the index is known, the value is fixed. For example,
\[ a_5=10. \]There is no uncertainty about the fifth term.
From a Sequence to a Random Sequence
Suppose a fair coin is tossed repeatedly. Define
\[ X_n= \begin{cases} 1, & \text{if toss } n \text{ is heads},\\ 0, & \text{if toss } n \text{ is tails}. \end{cases} \]Before the tosses occur, the values of
\[ X_1,X_2,X_3,\ldots \]are uncertain.
One possible realized sequence is
\[ 1,0,1,1,0,0,1,\ldots \]while another possible realization is
\[ 0,0,1,0,1,1,1,\ldots \]Probability describes how likely the different possible sequences are.
Formal Definition
A random sequence is an indexed collection of random variables
\[ \boxed{\{X_n:n\in I\}}, \]where the index set \(I\) is discrete.
Typical index sets include
\[ I=\{0,1,2,\ldots\}, \]or
\[ I=\{1,2,\ldots,N\}. \]Because the index changes in discrete steps, a random sequence is also called a discrete-time stochastic process.
Each Term Is a Random Variable
The notation \(X_n\) does not usually mean a fixed number before the random experiment is performed. It means a random variable associated with index \(n\).
For example,
\[ X_3 \]might represent the number of customers arriving during the third minute, the temperature measured on the third day, or the condition of a machine after its third operating cycle.
Every \(X_n\) has its own probability distribution.
Outcomes and Realized Values
Let \(\Omega\) be the sample space of the complete random experiment. For a particular outcome \(\omega\in\Omega\), each random variable \(X_n\) produces an ordinary value
\[ X_n(\omega). \]Thus the abstract random sequence
\[ X_0,X_1,X_2,\ldots \]becomes the realized numerical sequence
\[ X_0(\omega),X_1(\omega),X_2(\omega),\ldots. \]Sample Paths
A realized sequence
\[ \boxed{x_0,x_1,x_2,\ldots} \]generated by one outcome of the random experiment is called a sample path, trajectory, or realization.
The random sequence describes all possible paths together with their probabilities. A sample path is only one path that actually occurred.
Random Sequence Versus Sample Path
It is important to distinguish the random variables from their realized values.
Before observation:
\[ X_1,X_2,X_3,\ldots \]are random variables.
After one experiment:
\[ x_1,x_2,x_3,\ldots \]are observed values.
Uppercase letters are commonly used for random variables, while lowercase letters are commonly used for realized values.
Finite Random Sequences
A random sequence can contain only finitely many terms.
For example,
\[ X_1,X_2,\ldots,X_{12} \]might represent monthly sales for one year.
The entire sequence can then be treated as a random vector:
\[ \boxed{(X_1,X_2,\ldots,X_{12})}. \]Infinite Random Sequences
Many stochastic models use an indefinitely continuing sequence:
\[ X_0,X_1,X_2,\ldots. \]This does not mean that infinitely many observations must actually be collected. It means the mathematical model specifies how the process could continue for every finite index.
The Index Has an Interpretation
The index \(n\) often represents time, but it does not have to.
It may represent:
- the number of a repeated trial;
- a day, week, month, or year;
- a step in an algorithm;
- a generation in a population model;
- a position along a line or grid;
- an inspection cycle in a reliability system;
- the number of transitions completed by a system.
What matters is that the indices form an ordered discrete set.
The Values Need Not Be Numbers
The values of a random sequence can be numerical, but they can also be categorical or structural.
For example,
\[ X_n\in\{\text{working},\text{failed}\} \]could describe the condition of a machine after cycle \(n\).
A later lesson will formalize the collection of possible values as the state space.
Discrete-Valued Random Sequences
If each \(X_n\) takes values in a finite or countable set, the sequence is discrete-valued.
Examples include
\[ X_n\in\{0,1\}, \] \[ X_n\in\{0,1,2,\ldots\}, \]or
\[ X_n\in\{A,B,C\}. \]Counts, categories, success indicators, and system states commonly produce discrete-valued sequences.
Continuous-Valued Random Sequences
A random sequence may instead have real-valued terms.
For example,
\[ X_n\in\mathbb{R} \]could represent a sensor reading at the end of minute \(n\).
The time index is still discrete even though the observed values are continuous.
Discrete Time and Discrete Values Are Different Ideas
The phrase discrete-time describes the index set, not the set of possible values.
For example, measurements recorded once each hour form a discrete-time process even if each measurement can take any real value.
Thus a random sequence may be:
- discrete-time and discrete-valued; or
- discrete-time and continuous-valued.
Marginal Distributions
Each individual random variable \(X_n\) has a probability distribution called its marginal distribution.
For a discrete-valued term, we may write
\[ P(X_n=x). \]For example, if every toss in a fair-coin sequence is represented by \(0\) or \(1\), then
\[ P(X_n=1)=\frac12 \]for every \(n\).
Joint Distributions
Marginal distributions describe one position at a time. To describe how several positions behave together, we need a joint distribution.
For example,
\[ P(X_1=x_1,X_2=x_2,X_3=x_3) \]describes the probability that the first three terms take specified values simultaneously.
Joint distributions are essential because two random sequences can have the same one-term distributions but very different dependence across time.
Example: Same Marginals, Different Dependence
Let \(Y\) be a fair Bernoulli random variable:
\[ P(Y=0)=P(Y=1)=\frac12. \]Define
\[ X_n=Y \]for every \(n\).
Then each \(X_n\) individually satisfies
\[ P(X_n=0)=P(X_n=1)=\frac12. \]But once the first term is known, every later term is known because
\[ X_1=X_2=X_3=\cdots. \]The terms have identical marginal distributions but are maximally dependent.
Independent Terms
Random variables \(X_1,\ldots,X_k\) are independent when knowing some of them does not change the probability distribution of the others.
For discrete values, independence implies
\[ \boxed{ P(X_1=x_1,\ldots,X_k=x_k) = \prod_{i=1}^{k}P(X_i=x_i) }. \]A random sequence is called an independent sequence when every finite collection of distinct terms is independent.
Example: Independent Coin Tosses
For independent fair coin tosses represented by \(0\) and \(1\),
\[ P(X_n=0)=P(X_n=1)=\frac12. \]The probability of observing the first four values
\[ 1,0,1,1 \]is
\[ \left(\frac12\right)^4=\frac1{16}. \]Independence allows the joint probability to be found by multiplication.
Identically Distributed Terms
Random variables are identically distributed when they have the same probability distribution.
Thus
\[ X_1,X_2,X_3,\ldots \]are identically distributed if every position follows the same distributional rule.
Identically distributed does not automatically mean independent.
Independent and Identically Distributed Sequences
A sequence whose terms are both independent and identically distributed is called an i.i.d. sequence.
This is abbreviated
\[ \boxed{\text{i.i.d.}} \]Repeated independent tosses of the same coin are a standard example.
The two requirements are separate:
- independent describes the relationship among terms;
- identically distributed describes their individual distributions.
Bernoulli Random Sequences
A common random sequence consists of Bernoulli random variables.
If
\[ X_n= \begin{cases} 1, & \text{success},\\ 0, & \text{failure}, \end{cases} \]with
\[ P(X_n=1)=p \]and
\[ P(X_n=0)=1-p, \]then \(X_n\) has a Bernoulli distribution.
If all the terms are independent and use the same \(p\), the sequence is an i.i.d. Bernoulli sequence.
Counting Successes from a Random Sequence
A random sequence can be transformed into another sequence.
For a Bernoulli sequence, define
\[ S_n=X_1+X_2+\cdots+X_n. \]Then \(S_n\) counts the number of successes among the first \(n\) trials.
If the \(X_i\) are independent Bernoulli random variables with success probability \(p\), then
\[ S_n\sim\operatorname{Binomial}(n,p). \]This shows how a familiar probability distribution can arise from an underlying random sequence.
Categorical Random Sequences
A random sequence can record more than two categories.
For example,
\[ X_n\in\{\text{low},\text{medium},\text{high}\} \]might record a demand level during period \(n\).
If the probabilities are
\[ P(X_n=\text{low})=0.2, \] \[ P(X_n=\text{medium})=0.5, \] \[ P(X_n=\text{high})=0.3, \]then each term follows the same categorical distribution when these probabilities do not change with \(n\).
Count-Valued Random Sequences
Many random sequences record nonnegative counts.
Examples include:
- customers arriving during each minute;
- defects found in each production batch;
- emails received during each hour;
- failures observed during each testing cycle.
Such sequences often take values in
\[ \{0,1,2,\ldots\}. \]Real-Valued Measurement Sequences
Suppose a sensor records temperature once every minute. We may define
\[ X_n=\text{temperature at minute }n. \]Each \(X_n\) may take real values.
The sequence
\[ X_0,X_1,X_2,\ldots \]is random because future measurements are uncertain, even though the measurement times are fixed.
The Mean Sequence
If each \(X_n\) has a finite expectation, define
\[ \boxed{\mu_n=E[X_n]}. \]The values
\[ \mu_0,\mu_1,\mu_2,\ldots \]form the mean sequence.
The mean may remain constant or change with \(n\).
Example: A Changing Mean
Suppose
\[ X_n=n+Z_n, \]where
\[ E[Z_n]=0. \]Then
\[ E[X_n]=E[n+Z_n]=n. \]Therefore,
\[ \boxed{\mu_n=n}. \]The expected level of the sequence increases with the index.
The Variance Sequence
If the second moments exist, define
\[ \boxed{\sigma_n^2=\operatorname{Var}(X_n)}. \]The variance sequence records how much uncertainty appears at each position.
A constant variance does not imply that the terms are independent.
Covariance Across Positions
To measure linear dependence between two positions, we can use covariance:
\[ \boxed{ \operatorname{Cov}(X_m,X_n) = E[(X_m-E[X_m])(X_n-E[X_n])] }. \]If
\[ \operatorname{Cov}(X_m,X_n)=0, \]the variables are uncorrelated.
Independence implies zero covariance when the required moments exist, but zero covariance does not generally imply independence.
Autocovariance
When covariance is studied within one random sequence, it is often called autocovariance.
We may write
\[ \gamma(m,n)=\operatorname{Cov}(X_m,X_n). \]This function helps describe how values at different indices move together.
Strong positive covariance between nearby terms can indicate persistence, while negative covariance can indicate alternating behavior.
A Simple Dependent Sequence
Let
\[ Z_0,Z_1,Z_2,\ldots \]be independent random variables with
\[ E[Z_n]=0. \]Define
\[ X_n=Z_n+Z_{n-1} \]for \(n\ge 1\).
Then neighboring terms share one random component:
\[ X_n=Z_n+Z_{n-1}, \] \[ X_{n+1}=Z_{n+1}+Z_n. \]Because both contain \(Z_n\), the sequence is generally dependent.
Dependence Can Carry Information Forward
In many stochastic systems, the current value provides information about the future.
For example, if a machine is currently degraded, it may be more likely to be degraded or failed on the next inspection.
This kind of dependence is central to later models such as Markov chains, queues, reliability processes, and random walks.
Transforming a Random Sequence
A function can be applied to every term of a random sequence.
If
\[ Y_n=g(X_n), \]then
\[ Y_0,Y_1,Y_2,\ldots \]is another random sequence.
For example, if \(X_n\) is a temperature measurement, then
\[ Y_n=\mathbf{1}_{\{X_n>30\}} \]produces a sequence that records whether each measurement exceeds \(30\).
Indicator Sequences
An indicator random variable equals \(1\) when an event occurs and \(0\) otherwise.
For events \(A_n\), define
\[ I_n= \begin{cases} 1, & A_n\text{ occurs},\\ 0, & A_n\text{ does not occur}. \end{cases} \]Then
\[ I_1,I_2,I_3,\ldots \]is an indicator sequence.
Indicator sequences are useful for turning qualitative events into numerical data that can be counted and averaged.
Cumulative Sequences
Given a random sequence \(X_1,X_2,\ldots\), define cumulative sums by
\[ S_n=\sum_{k=1}^{n}X_k. \]Then
\[ S_1,S_2,S_3,\ldots \]is itself a random sequence.
Cumulative sequences convert step-by-step changes into total accumulated change.
This construction will reappear in random walks and counting processes.
Example: Inventory Changes
Suppose \(X_n\) is the net inventory change during period \(n\). Positive values represent net additions and negative values represent net withdrawals.
If the initial inventory is \(S_0\), then
\[ S_n=S_0+\sum_{k=1}^{n}X_k. \]The sequence
\[ S_0,S_1,S_2,\ldots \]records the evolving inventory level.
The random increments \(X_n\) generate the random state sequence \(S_n\).
Stationarity as a Preview
Some random sequences have probability behavior that does not change when the index is shifted.
Very roughly, a sequence is called stationary when its distributional structure is stable over time.
For example, a stationary sequence may have
\[ E[X_n]=\mu \]for every \(n\), although equal means alone are not enough to establish stationarity.
Stationarity is an important concept in stochastic modeling, but it is stronger than simply saying that all terms have the same expected value.
Random Sequences as Models
A random sequence is not the physical system itself. It is a mathematical representation of uncertainty in that system.
A useful model requires decisions about:
- what the index means;
- what each random variable measures;
- what values are possible;
- how the marginal distributions are specified;
- how terms depend on one another;
- which assumptions are reasonable for the application.
Modeling Assumption: Independence
Independence is mathematically convenient, but it should not be assumed automatically.
Daily temperatures, financial prices, machine conditions, and queue lengths often show dependence across nearby times.
Repeated randomized trials may be approximately independent, but physical or behavioral systems often retain memory.
The dependence structure is part of the model and should be justified.
Modeling Assumption: Identical Distributions
Identical distributions are also a modeling choice.
A sequence of sales values may have different distributions in different seasons. A machine may become more failure-prone as it ages. A population may face changing environmental conditions.
If the distribution of \(X_n\) changes with \(n\), the sequence is not identically distributed.
Example: Independent but Not Identically Distributed
Suppose \(X_n\) are independent Bernoulli random variables with
\[ P(X_n=1)=\frac{1}{n+1}. \]The variables can be independent, but their success probabilities change with \(n\).
Therefore, the sequence is independent but not identically distributed.
Example: Identically Distributed but Not Independent
Let \(Y\) be Bernoulli with
\[ P(Y=1)=p. \]Set
\[ X_n=Y \]for every \(n\).
Every \(X_n\) has the same Bernoulli distribution, but all the terms are tied to the same random variable. They are identically distributed but not independent.
Random Sequences and Conditional Probability
Dependence is often described through conditional probabilities.
For example,
\[ P(X_{n+1}=j\mid X_n=i) \]describes the probability of observing value \(j\) at the next index given that the current value is \(i\).
This type of transition probability becomes central in the study of Markov chains.
Random Sequences and State Updates
Many models can be expressed through a recursive update:
\[ X_{n+1}=F(X_n,\xi_{n+1}), \]where \(\xi_{n+1}\) represents new random input arriving at the next step.
This equation separates two ingredients:
- the current state \(X_n\);
- new randomness \(\xi_{n+1}\).
Recursive stochastic models provide a natural bridge from deterministic recurrences to stochastic processes.
Example: Random Recurrence
Suppose
\[ X_{n+1}=X_n+\xi_{n+1}, \]where each \(\xi_n\) equals \(1\) or \(-1\).
Then every next value is obtained by taking the current value and adding a random step.
Repeated application gives
\[ X_n=X_0+\xi_1+\xi_2+\cdots+\xi_n. \]This is the basic structure of a random walk, which will be developed later in this section.
One-Step and Multi-Step Questions
Random-sequence problems often ask questions at different horizons.
A one-step question might ask
\[ P(X_{n+1}=j\mid X_n=i). \]A multi-step question might ask
\[ P(X_{n+k}=j\mid X_n=i). \]The distinction matters because uncertainty accumulates as the sequence evolves.
Observed Data from a Random Sequence
If we observe
\[ x_1,x_2,\ldots,x_N, \]we have one finite sample path from the underlying random sequence.
We can compute summaries such as the sample average
\[ \bar{x} = \frac1N\sum_{n=1}^{N}x_n. \]But a single observed path does not automatically reveal the entire probability law of the sequence.
Ensemble View and Time View
There are two useful ways to think about a random sequence.
The ensemble view compares many possible outcomes at a fixed index \(n\).
The time view follows one realized path across many indices.
These perspectives are related but not automatically interchangeable. Moving from one to the other requires additional assumptions.
Simulating a Random Sequence
Simulation produces sample paths from a specified probability model.
For an i.i.d. Bernoulli sequence with success probability \(p\), a simulation can follow this procedure:
- Choose the number of terms \(N\).
- For each \(n=1,\ldots,N\), generate an independent random number \(U_n\) between \(0\) and \(1\).
- Set \(X_n=1\) if \(U_n\le p\).
- Otherwise set \(X_n=0\).
- Record the resulting sequence.
Example: Simulating Ten Bernoulli Trials
Let
\[ p=0.4. \]One simulated path might be
\[ 0,1,0,0,1,1,0,0,0,1. \]The number of successes in this particular path is
\[ 4. \]Another simulation will usually produce a different sequence.
Simulation generates possible behavior; it does not replace the underlying probability model.
Simulation and Long-Run Behavior
For a long i.i.d. Bernoulli sequence with success probability \(p\), the fraction of successes
\[ \frac{X_1+\cdots+X_n}{n} \]tends to remain increasingly close to \(p\) as \(n\) becomes large.
This connects random sequences to the law of large numbers studied earlier.
Individual paths fluctuate, but aggregate behavior can become stable.
Randomness Does Not Mean Patternless
A random sequence may have strong statistical structure.
For example, its terms may:
- share a common distribution;
- show persistence or alternation;
- have a long-run average;
- obey a transition rule;
- accumulate through random increments;
- approach a stable distribution.
Stochastic modeling is the study of this probabilistic structure.
Common Error: Confusing a Random Variable with Its Value
Writing
\[ X_3=7 \]after observing the third term is fine.
But before the experiment, \(X_3\) represents a random variable, not a number known in advance.
The distinction between \(X_3\) and a realized value \(x_3\) helps separate the probability model from the observed data.
Common Error: Assuming Independence
A sequence notation such as
\[ X_1,X_2,X_3,\ldots \]does not imply independence.
Dependence must be stated, derived, or justified from the model.
In particular,
\[ P(X_2=x_2\mid X_1=x_1) \]may differ from
\[ P(X_2=x_2). \]Common Error: Assuming Equal Distributions
Terms in the same random sequence do not have to share the same distribution.
It is possible that
\[ X_1\sim\operatorname{Bernoulli}(0.1) \]while
\[ X_{10}\sim\operatorname{Bernoulli}(0.6). \]The index itself may influence the probability law.
Common Error: Confusing Discrete Time with Discrete Values
A sequence indexed by
\[ 0,1,2,\ldots \]has discrete time.
Its values may still be continuous, such as temperatures, prices, positions, or voltages.
Always ask separately what the index set is and what values each random variable may take.
Common Error: Treating One Path as the Entire Process
The observed sequence
\[ 2,1,3,0,2,\ldots \]is one realization.
The stochastic model also includes other paths that could have occurred and the probabilities assigned to them.
A sample path is evidence generated by the model, not the model itself.
Common Error: Using Marginals to Infer Dependence
Knowing
\[ P(X_n=x) \]for every \(n\) does not determine how different terms depend on one another.
Dependence requires joint or conditional probability information.
Two sequences can have identical marginal distributions and completely different joint behavior.
A Reliable Procedure for Reading a Random-Sequence Model
- Identify the index set.
- State what \(X_n\) represents.
- Identify the possible values of \(X_n\).
- Determine the marginal distribution of each term.
- Check whether the distributions change with \(n\).
- Determine whether the terms are independent or dependent.
- Use joint or conditional probabilities when dependence matters.
- Separate the random sequence from any one observed sample path.
- Compute means, variances, or covariances when useful.
- Use simulation when a sample path or empirical approximation is needed.
Example: Full Model Description
Suppose a component is tested once per hour. Define
\[ X_n= \begin{cases} 1, & \text{component passes test }n,\\ 0, & \text{component fails test }n. \end{cases} \]Assume
\[ P(X_n=1)=0.98 \]for every \(n\), and assume the tests are independent.
Then:
- the index set is discrete;
- the value set is \(\{0,1\}\);
- each term is Bernoulli with parameter \(0.98\);
- the terms are independent;
- the sequence is i.i.d.
Example: A Dependent Condition Sequence
Suppose a machine is classified after each inspection as
\[ X_n\in\{\text{good},\text{degraded},\text{failed}\}. \]The probability of the next condition may depend strongly on the current condition.
For instance,
\[ P(X_{n+1}=\text{failed}\mid X_n=\text{degraded}) \]may be much larger than
\[ P(X_{n+1}=\text{failed}\mid X_n=\text{good}). \]This is a random sequence with meaningful dependence across indices.
Practice 1: Identify the Random Sequence
A factory records the number of defective items in each production batch. Let \(X_n\) be the number of defects in batch \(n\).
Is \(\{X_n\}\) a random sequence? What does the index \(n\) represent?
Practice 2: Sample Path
A Bernoulli random sequence is observed for eight trials and produces
\[ 1,0,0,1,1,0,1,0. \]Is this list the random sequence itself or a sample path from the random sequence?
Practice 3: Discrete Time and Continuous Values
A sensor records air pressure once every five minutes.
Is this a discrete-time sequence? Are the values necessarily discrete?
Practice 4: Independence
Suppose \(X_1\) and \(X_2\) are independent, with
\[ P(X_1=1)=0.6 \]and
\[ P(X_2=1)=0.5. \]Find
\[ P(X_1=1,X_2=1). \]Practice 5: Identically Distributed?
Suppose
\[ X_n\sim\operatorname{Bernoulli}\left(\frac{n}{n+1}\right). \]Are the variables identically distributed?
Practice 6: i.i.d. Sequence
A fair die is rolled repeatedly, and \(X_n\) records the result of roll \(n\). Assume the rolls are independent.
Explain why \(\{X_n\}\) is i.i.d.
Practice 7: Mean Sequence
Suppose
\[ X_n=2n+Z_n \]and
\[ E[Z_n]=0. \]Find \(E[X_n]\).
Practice 8: Cumulative Count
Let \(X_n\) indicate whether a customer arrives during interval \(n\):
\[ X_n\in\{0,1\}. \]Write an expression for the total number of arrivals during the first \(20\) intervals.
Practice 9: Dependence
Let \(Y\) be a Bernoulli random variable and define
\[ X_n=Y \]for every \(n\).
Are the terms identically distributed? Are they independent?
Practice 10: Random Recurrence
Suppose
\[ X_{n+1}=X_n+\xi_{n+1}, \]where each \(\xi_n\) equals \(1\) or \(-1\).
Express \(X_3\) in terms of \(X_0,\xi_1,\xi_2,\xi_3\).
Practice Answers
Practice 1
Yes. Each \(X_n\) is an uncertain count, and the index \(n\) identifies the production batch.
Practice 2
The list
\[ 1,0,0,1,1,0,1,0 \]is one realized sample path from the underlying random sequence.
Practice 3
Yes. The observations occur at discrete five-minute indices. The measured pressure values can still be continuous.
Practice 4
Independence gives
\[ P(X_1=1,X_2=1) = P(X_1=1)P(X_2=1). \]Therefore,
\[ \boxed{P(X_1=1,X_2=1)=0.6(0.5)=0.3}. \]Practice 5
No. The Bernoulli success probability
\[ \frac{n}{n+1} \]changes with \(n\), so the distributions are not identical.
Practice 6
Each roll has the same discrete uniform distribution on
\[ \{1,2,3,4,5,6\}, \]and the rolls are independent. Therefore the sequence is i.i.d.
Practice 7
\[ E[X_n] = E[2n+Z_n] = 2n+E[Z_n] = \boxed{2n}. \]Practice 8
The total number of arrivals is
\[ \boxed{S_{20}=\sum_{n=1}^{20}X_n}. \]Practice 9
The terms are identically distributed because each one equals the same Bernoulli random variable \(Y\). They are not independent because observing one term reveals every other term.
Practice 10
Apply the recurrence repeatedly:
\[ X_1=X_0+\xi_1, \] \[ X_2=X_0+\xi_1+\xi_2, \]so
\[ \boxed{X_3=X_0+\xi_1+\xi_2+\xi_3}. \]What to Remember
A random sequence is a discrete-indexed collection of random variables:
\[ \boxed{\{X_n\}_{n\ge 0}}. \]- Each \(X_n\) is a random variable.
- A realized numerical sequence is a sample path.
- The index is discrete, but the values may be discrete or continuous.
- Marginal distributions describe individual terms.
- Joint and conditional distributions describe dependence across terms.
- Independent and identically distributed are separate properties.
- An i.i.d. sequence has both properties.
- Means, variances, and covariances summarize important features of the sequence.
- Cumulative sums and recursive updates create new stochastic processes from random inputs.
The key conceptual shift is this: a stochastic process is not one uncertain number, but an organized family of uncertain quantities indexed by progression through a system.
Where We Go Next
A random sequence tells us that a system can occupy uncertain values at discrete indices. To model such systems precisely, we need to specify the collection of values the system is allowed to occupy.
The next lesson, State Spaces, develops that idea and shows how finite, countable, continuous, and structured state spaces provide the setting in which stochastic processes evolve.