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Probability Unit 6: Continuous Random Variables

Continuous random variables take values in intervals, and their probabilities are areas under a density curve. This unit replaces sums with integrals and adds tools such as percentiles and transformations.

Covers: Probability density functions, CDFs and survival functions, Expected value and variance by integration, Medians, percentiles and modes, Transformations of a random variable, Mixed distributions, Insurance payments with continuous losses

Study guide

A continuous random variable X has a probability density function f with f(x) ≥ 0 and total area ∫ f(x) dx = 1. Probabilities are areas: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx. As a result, P(X = a) = 0 for any single value, so it does not matter whether an inequality is strict. A density is not a probability and can exceed 1. To find an unknown constant in a density, integrate the unnormalized function over its support and divide 1 by the result.

The CDF F(x) = P(X ≤ x) = ∫ from −∞ to x of f(t) dt is continuous and nondecreasing, and f(x) = F′(x) wherever the derivative exists. The survival function S(x) = 1 − F(x) = P(X > x) is common in insurance problems. Percentiles come from the CDF: the p-th percentile solves F(x) = p, and the median solves F(m) = ½. The mode is where the density peaks, found by setting f′(x) = 0. Conditional probabilities work as before, for example P(X > a | X > b) = S(a)/S(b) when a > b.

Expectations become integrals. E[X] = ∫ x f(x) dx, E[g(X)] = ∫ g(x) f(x) dx, and Var(X) = E[X²] − (E[X])², with the same linearity and scaling rules as in the discrete case. For a nonnegative variable, E[X] = ∫₀^∞ S(x) dx, which is often easier than finding f. A valid density can still have an infinite mean: f(x) = 1/x² on [1, ∞) integrates to 1, but ∫ x f(x) dx diverges. Always check that an expectation integral converges.

To find the distribution of Y = g(X), use the CDF method: write F_Y(y) = P(g(X) ≤ y), rewrite the event in terms of X, use F_X, and differentiate. For an increasing g this gives f_Y(y) = f_X(g⁻¹(y)) × |d g⁻¹(y)/dy|. Some variables are mixed, with a point mass and a continuous part; a loss that is 0 with probability 0.3 is an example. Compute their expectations by adding the discrete and continuous contributions. Insurance payments such as max(X − d, 0) or min(X, u) are functions of X, so their expectations are integrals of the payment against the density, or integrals of the survival function.

Key ideas

• Probabilities are areas under the density, and single points have probability 0.
• A density can exceed 1; only its total area must be 1.
• f = F′, and percentiles solve F(x) = p.
• Expectations are integrals, with the same linearity rules as sums.
• For nonnegative X, E[X] = ∫₀^∞ S(x) dx.
• Use the CDF method to find the distribution of a function of X.
• Heavy-tailed densities can have infinite means.

Formulas and rules

Common mistakes

• Treating f(x) as a probability or expecting f(x) ≤ 1.
• Forgetting to integrate over the whole support when finding a normalizing constant.
• Using g(E[X]) in place of E[g(X)].
• Substituting y into the density without the Jacobian factor in a transformation.
• Forgetting the point mass in a mixed distribution.

Before the test, make sure you can

• Find the constant that makes a function a valid density.
• Compute probabilities by integrating a density or subtracting CDF values.
• Derive the density from a CDF and the CDF from a density.
• Calculate E[X], E[g(X)] and Var(X) by integration.
• Find medians, percentiles and modes.
• Find the distribution of Y = g(X) with the CDF method.
• Compute expected insurance payments using the survival function.
• Handle mixed distributions with a point mass.

Words to know

Continuous random variable

A random variable with a density, taking values in an interval, with P(X = a) = 0 for every a.

Probability density function

f(x) ≥ 0 with total area 1; probabilities are areas under f.

CDF

F(x) = ∫ from −∞ to x of f(t) dt = P(X ≤ x).

Survival function

S(x) = P(X > x) = 1 − F(x).

Percentile

The value x with F(x) = p for the p-th quantile.

Median

The value m with F(m) = ½.

Mode

The value at which the density is largest.

Expected value (continuous)

E[X] = ∫ x f(x) dx.

LOTUS (continuous)

E[g(X)] = ∫ g(x) f(x) dx.

Survival-function mean

For X ≥ 0, E[X] = ∫₀^∞ S(x) dx.

CDF method

Find the distribution of Y = g(X) by computing P(g(X) ≤ y) and differentiating.

Jacobian factor

The term |dx/dy| that appears in the density of a transformed variable.

Mixed distribution

A distribution with both point masses and a continuous part.

Support

The set of values where the density is positive.

Heavy tail

A slowly decaying density, which can make moments such as the mean infinite.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.