AcademyProbability
Academy
Continuous Random Variables
Level 1 - Math II (Physics) topic page in Probability.
Principle
A continuous random variable has probabilities described by area under a density curve. Individual exact values have probability zero; intervals have probability equal to integrated density.
Notation
\(X\)
continuous random variable
\(x\)
a possible measured value of X
\(f_X(x)\)
probability density function, or PDF
\(F_X(x)\)
cumulative distribution function, or CDF
\(D_X\)
support where the density may be nonzero
\(a,b\)
interval endpoints with a\le b
\([f_X]\)
density unit, the inverse unit of X
Method
Step 1: Identify the support and density unit
The support states where the variable can lie. If \(X\) is measured in seconds, then \(f_X(x)\) is measured in per second, so density times interval length is dimensionless probability.
Step 2: Integrate density over intervals
Continuous probability is area. The endpoints do not change the probability because a single point has probability zero.
Interval probability
\[P(a\le X\le b)=\int_a^b f_X(x)\,dx\]
CDF definition
\[F_X(x)=P(X\le x)\]
CDF as accumulated area
\[F_X(x)=\int_{-\infty}^{x} f_X(t)\,dt\]
Density from CDF where differentiable
\[F_X'(x)=f_X(x)\]
Point probability
\[P(X=x)=\int_x^x f_X(t)\,dt=0\]
Rules
Density total area
\[\int_{-\infty}^{\infty}f_X(x)\,dx=1\]
Interval probability
\[P(a\le X\le b)=\int_a^b f_X(x)\,dx\]
CDF
\[F_X(x)=P(X\le x)\]
CDF derivative where differentiable
\[F_X'(x)=f_X(x)\]
Point probability
\[P(X=x)=0\]
Examples
Question
A value is uniformly distributed on
\[[2,6]\]
Find the density and \[P(3\le X\le5)\]
Answer
The support length is
\[4\]
so \[f_X(x)=1/4\]
on \[[2,6]\]
Then \[P(3\le X\le5)=\int_3^5\frac{1}{4}\,dx=\frac{5-3}{4}=\frac{1}{2}.\]
Checks
- A density value can exceed \(1\); it is not itself a probability.
- Integrated area, not height, must be between \(0\) and \(1\).
- The total area under a valid PDF must be \(1\).
- The density unit is the inverse of the unit of the random variable.
- For continuous variables, \(P(X=x)=0\) even when \(x\) lies in the support.