Calculus I · Unit 3B · lesson
Probability Density and Expected Value
Learn Probability Density and Expected Value through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Physics and quantitative applicationsWhat this section is building
Learn Probability Density and Expected Value through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
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.
Draw a coordinate system, define the slice at a general position, express every changing factor in one variable, and state the domain and units.
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.
Learning objectives
Interpret probability as accumulated density and calculate expected value for a continuous model.
Probability Density and Expected Value
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 measures the probability that the random variable falls between and .
Expected value is a weighted average in which each possible value is weighted by its density. The integral 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 satisfies and
Probabilities are areas:
The expected value is
when the improper integral converges.
A simple triangular density
Let on and elsewhere. It is normalized because
Then
and
u3b-prob-01For density on , find .
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Integrate the density from 0 to 1/2.
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