In the study of statistics and probability, we often begin by looking at static events—the probability of a single coin flip or the result of a one-time draw from a deck of cards. However, the real world is rarely that simple. Most phenomena, from the fluctuating prices on the Nairobi Securities Exchange to the arrival of patients at a hospital, happen over time. This is the domain of the Stochastic Process. It is the mathematical language of systems that evolve randomly, and for many students, it is the most challenging unit in the actuarial and statistical sciences.
Below is the exam paper download link
PDF Past Paper On Stochastic Process For Revision
Above is the exam paper download link
To help you move from being overwhelmed by notation to mastering the flow of random variables, we have compiled a high-impact revision guide based on the most frequent hurdles found in recent examination sittings.
What exactly is a Stochastic Process?
A stochastic process is a collection of random variables indexed by time. Instead of looking at a single number, you are looking at a “path.” If you are tracking the number of customers in a supermarket throughout the day, the state of the system at 10:00 AM is a random variable, and the state at 11:00 AM is another. The goal is to understand how these variables relate to one another as time moves forward.
What is the ‘Markov Property’ and why is it so famous?
The Markov Property is the “Gold Standard” of stochastic modeling. It states that the future state of a process depends only on the current state, not on the sequence of events that preceded it. This is often called “memorylessness.” In an exam, you will likely be asked to prove if a process is a Markov Chain. If the probability of moving to state $J$ tomorrow only requires knowledge of state $I$ today, you have a Markovian system.
How do we use ‘Transition Probability Matrices’?
In discrete-time Markov Chains, we use a matrix to show the probability of moving from one state to another in a single step. Each row of the matrix must sum to 1, as the system must transition to some state. A common exam question involves “higher-order transitions,” where you are asked to find the probability of being in a certain state after three or four steps. This usually requires performing matrix multiplication ($P^n$).
What is a ‘Poisson Process’?
A Poisson Process is the most common model for “counting” events that happen at a constant average rate. Think of it as a clock that ticks at random intervals. For your revision, remember three key features:
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The number of events in non-overlapping time intervals is independent.
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The probability of two events happening at the exact same micro-second is zero.
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The time between arrivals (inter-arrival time) follows an Exponential Distribution.
What are ‘Martingales’ in Finance and Probability?
A Martingale is a model of a “fair game.” It is a stochastic process where the expected value of the next observation, given all past observations, is equal to the current value. There is no predictable trend or “drift.” Martingales are essential for understanding risk-neutral pricing in actuarial science and are a favorite topic for advanced theory questions in past papers.
Can you explain ‘Random Walks’?
A Random Walk is a process where an object moves in steps, each step being a random variable. The most famous example is the “Drunkard’s Walk.” In a 1D random walk, you move left or right with certain probabilities. In your revision, pay close attention to the Gambler’s Ruin problem, which uses random walk logic to predict when a player with finite wealth will eventually hit zero.
What is ‘Stationarity’ in a process?
A process is Stationary if its statistical properties (like mean and variance) do not change over time. If you take a “slice” of the data today and a slice ten years from now, they should look effectively the same. This is a critical assumption for many forecasting models, as it allows us to assume that the past is a reliable guide to the future.
Conclusion
Stochastic Process is a unit that rewards those who can visualize “motion” within mathematics. It isn’t just about memorizing formulas; it’s about understanding how randomness propagates through a system. The best way to build this intuition is to work through previous years’ questions, where you are forced to define state spaces and calculate transition limits manually.
To help you master these temporal patterns and ace your finals, we have provided a link to the essential PDF revision resource below.
Last updated on: March 24, 2026