Continuous Distributions

The Smooth Side of Chance

Discrete variables jump from 1 to 2, but Continuous variables flow. They can take any value— $1.5$ , $1.55$ , $1.5555...$ . This means the probability of any exact value is actually zero ( $P(X=1.75) = 0$ ). Instead, we talk about the probability of falling within a range $[a, b]$ , which is the area under the Probability Density Function (PDF), $f(x)$ .

✦Intuition

Area is Probability

For discrete distributions, we sum probabilities. For continuous distributions, we integrate the PDF over an interval. The total area under the curve from $-\infty$ to $\infty$ must exactly equal 1.

Summary of Continuous Models

From simple flat lines to unpredictable heavy tails, we will explore the following core continuous distributions:

Distribution	Core Concept	Mean (E[X])	Variance (Var(X))
Uniform	All outcomes equally likely in $[a, b]$	$(a+b)/2$	$(b-a)^2/12$
Normal	The ubiquitous bell curve	$\mu$	$\sigma^2$
Exponential	Time between independent events	$1/\lambda$	$1/\lambda^2$
Cauchy	Heavy tails, unpredictable extremes	Undefined	Undefined

Let's begin by exploring the most straightforward continuous model: the Uniform Distribution.

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The Poisson Distribution

Course Progression

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The Uniform Distribution