In this episode we introduce moments, the bedrock of studying random variables and risk.
We begin with a concise overview, then zero in on simple (raw) moments, the foundation upon which central and standardized moments are built. You’ll learn how to compute an expected value (the first moment) and how to extend the same principles to higher-order simple moments.
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In this episode, we dive into the critical difference between extremes and outliers. As we will soon discover, understanding this distinction is vital for accurate modeling, especially when assessing (tail) risk.
We'll explore how both concepts relate to the range of possible values--the support--of a random variable and show you how to avoid common modeling pitfalls. 
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In this episode, building on the cumulative distribution function F (CDF) introduced in Episode 15, we define the quantile function Q, the survival function S, and the probability density function f (PDF). 
We then explore how each of these tools helps describe the behavior of a random variable.
Along the way, we also introduce Value-at-Risk (VaR), which becomes quite straightforward once you understand quantiles.
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In this episode, we continue laying the foundations of our mathematical treatment of risk. In particular, we introduce the cumulative distribution function, or CDF, as a tool for describing the probabilistic behavior of a random variable. We illustrate this with a simple example and discuss the basic characteristics of this interesting non-decreasing function, which will soon help us address very practical matters.
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In this episode, we answer the question we left hanging at the end of Episode 13: What is a sigma-algebra?
We will explore the properties of a sigma-algebra and explain why we want our event space to be one. Next, we will discuss probability spaces in a simple yet rigorous way, which will set the stage for understanding random variables and the mathematical treatment of risk measures.
Finally, we will discover why it can be inconvenient to swim naked.
Are you ready to wear a swimsuit?


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In this episode, we come back to the definition of risk and start discussing how we can measure it. That is why we need to introduce risk measures. But that requires us to be more formal and speak about the sample space and the event space in a more mathematical way.
Ready to discover the power set?
We are paving the way to introduce the probability space, a pivotal concept for us.
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In this episode, we conclude our overview of the various schools of thought on probability.
We begin by examining the logicist perspectives of Keynes, Jeffreys, and Jaynes before transitioning to Kolmogorov's axiomatic definition.
We will briefly review the three basic axioms of probability and explain why the axiomatic approach is not a definition in the traditional sense, but rather a framework that transcends the objectivist–subjectivist debate.
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This episode introduces logicism, examining how modern logic informs our understanding of probability. Can logicism be considered as a bridge between probabilistic objectivism and subjectivism? 
What did Wittgenstein think about probability?
Is probability the new logic of science?


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In this episode, we delve into subjective probability by drawing on Bruno de Finetti’s ideas. We show how viewing probability as a fair bet may lead to coherent estimates. We discuss how de Finetti challenges the circular and limiting nature of the classical and frequentist definitions, arguing that personal judgment, informed by data, experts, and even intuition, is key.
We also covers advanced ideas like Jeffrey's radical probabilism, ultimately emphasizing the need for continuous learning and humility in decision-making. 
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We introduce the subjectivist perspective on probability, featuring thinkers like Bruno de Finetti, Frank Ramsey, and Leonard Savage, who argue that probability is not an objective entity but rather something we invent to manage uncertainty arising from our limited knowledge.
De Finetti’s “betting” approach interprets probability as the price one would pay for a chance to win a fixed payoff, showing how different beliefs produce different numerical values.
From Laplace’s demon to everyday life, the notion of chance emerges because our knowledge is partial, compelling us to create probability and risk rather than discover them as objective realities. 
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