Continuous Random Variables and Distributions

Summary

The whole chapter in one scroll — read this first, then dive into the detail below.

• A continuous random variable has infinitely many possible values, so we use a probability density function, f(x), instead of a probability mass function.
• Two rules make something a valid p.d.f: it's never negative, and the total area under it equals 1.
• Probabilities are found by integrating f(x) between two limits — the area under the curve.
• The cumulative distribution function, F(x), is the "running total" of probability up to a point: F(x) = P(X ≤ x).
• Mean and variance for continuous variables use the same logic as discrete ones, just with integrals instead of sums.
• The mode is where the graph peaks; the median is where F(x) = 0.5; comparing them tells you the skew.
• The continuous uniform distribution U[a, b] is the "flat rectangle" special case — every value in the range is equally likely.

1 · What Even Is a Continuous Random Variable?

Setting the foundation before any formulas show up.

Think about height. If I asked "what's the probability someone is exactly 175.0000000... cm tall?" — the honest answer is basically zero, because height can take an infinite number of decimal values. There's always a "more precise" measurement hiding underneath. This is the core difference from something like rolling a die, where the outcomes are countable and separate (1, 2, 3, 4, 5, 6).

Because any single exact value has probability zero, we stop asking "what's the chance of exactly this value?" and instead ask "what's the chance the value falls in this range?" That's what the probability density function, f(x), is built for — it doesn't give you a probability directly, it gives you a "density" that you turn into a probability by finding an area underneath it.

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