Calculus I · Unit 3B · lesson

Probability Density and Expected Value

Concept

Learning objectives

Interpret probability as accumulated density and calculate expected value for a continuous model.

Probability Density and Expected Value

Explanation

Probability is continuous mass distributed over outcomes

For a continuous random variable, a density function does not give the probability of one exact value. Probabilities come from integrating density over intervals. The total area under a valid density must equal one, and the density must be nonnegative. Thus $\int_a^b f(x)\,dx$ measures the probability that the random variable falls between $a$ and $b$ .

Expected value is a weighted average in which each possible value is weighted by its density. The integral $E[X]=\int x f(x)\,dx$ is therefore closely related to center of mass. A density may be greater than one at some points without violating probability rules; only integrated probability must lie between zero and one. Always verify normalization before using a proposed function as a probability density.

A probability density $p(x)$ satisfies $p(x)\ge0$ and

\int_{-\infty}^{\infty}p(x)dx=1.

Probabilities are areas:

P(a\le X\le b)=\int_a^bp(x)dx.

The expected value is

E[X]=\int_{-\infty}^{\infty}xp(x)dx,

when the improper integral converges.

Worked example

A simple triangular density

Let $p(x)=2x$ on $[0,1]$ and $0$ elsewhere. It is normalized because

\int_0^12x\,dx=1.

Then

P(X\le1/2)=\int_0^{1/2}2x\,dx=\frac14,

and

E[X]=\int_0^1x(2x)dx=\frac23.

Interactive checku3b-prob-01

For density $p(x)=2x$ on $[0,1]$ , find $P(X\le1/2)$ .

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Integrate the density from 0 to 1/2.

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After the explanation

Use the section idea

Reading lens

Translate the situation into rate, density, force, pressure, or probability before calculating; the integral is the final accumulation step, not the first modeling decision.

Mental model

Work adds force through distance, pumping adds slice weight through lift distance, pressure adds depth-dependent strip force, and marginal or probability models add weighted local contributions.

Decision

Draw a coordinate system, define the slice at a general position, express every changing factor in one variable, and state the domain and units.

Common trap

Confusing mass density with weight density, measuring depth from the wrong reference, or integrating a marginal quantity without an initial value when a total function is requested.

Check yourself

Does the setup respond correctly when the slice moves, and can a units or scale check expose a missing factor?

Source & rights

Original instruction with traceable references.

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