Properties of a probability density
A density assigns probability through area. It must be non-negative on its support, have total area 1, and be zero outside that support
\[f(x)\ge0,\quad\int_a^b f(x)\,dx=1,\quad f(x)=0\text{ outside }[a,b]\]
- Check non-negativity The density must be at least zero at every point of its declared support \(\displaystyle f(x)\ge0\quad\text{for }a\le x\le b\)
- Check total area The area under the density across its complete support must be one \(\displaystyle \int_a^b f(x)\,dx=1\)
- State zero elsewhere A density is zero outside its declared support \(\displaystyle f(x)=0\quad\text{for }x<a\text{ or }x>b\)
- Normalise with the complete area Keep the unknown constant in the integral, set the total area equal to one and solve \(\displaystyle \int_0^2kx\,dx=2k=1\;\Rightarrow\;k=\frac{1}{2}\)
Worked example
Find the normalisation constant and check the density conditions for this proposed rule
\[f(x)=kx,\quad0\le x\le3,\quad f(x)=0\text{ elsewhere}\]
- \(\displaystyle \int_0^3kx\,dx=k\left[\frac{x^2}{2}\right]_0^3=\frac{9k}{2}\)Integrate the rule over the complete support and keep k as a factor
- \(\displaystyle \frac{9k}{2}=1\;\Rightarrow\;k=\frac{2}{9}\)Set the complete area equal to 1 and solve for the positive constant
- \(\displaystyle f(x)=\frac{2x}{9}\ge0\quad\text{for }0\le x\le3\)The normalised rule is non-negative throughout its support
- \(\displaystyle f(x)=0\quad\text{outside }[0,3]\)The zero-elsewhere statement supplies the remaining density condition
Common mistakes
Checking area without non-negativity
✕\(\displaystyle f(x)=3x-\frac{1}{2},\quad\int_0^1f(x)\,dx=1\;\Rightarrow\;\text{valid}\)✓\(\displaystyle f(0)=-\frac{1}{2}<0\;\Rightarrow\;\text{not valid}\)
A signed integral equal to 1 is not enough; the density must also be non-negative on its supportDropping the normalisation constant
✕\(\displaystyle \int_0^2kx\,dx=\int_0^2x\,dx=2\)✓\(\displaystyle \int_0^2kx\,dx=k\int_0^2x\,dx=2k\)
The constant multiplies the whole rule, so it remains in the area calculationKeeping a negative value for k
✕\(\displaystyle k=-\frac{1}{2}\)✓\(\displaystyle k=\frac{1}{2}\)
The chosen constant must keep the density non-negative on its support, so a negative value is rejectedDifficulty levels
introCheck a simple linear density on a bounded interval
standardNormalise a piecewise-linear density or identify one failed density condition
challengeNormalise an improper exponential or circular density and reject an invalid root
examChoose the valid or invalid density reason and calculate a normalisation consequence
Questions about this topic
Why must the total area under a density be 1?+
The total area represents the probability that the random variable lies somewhere in its support. That event has probability 1
Can a function with area 1 be a density if it is negative?+
No. Probability cannot be negative, so the density must be non-negative at every point of its support as well as having total area 1
Why is the constant kept inside the integral when normalising?+
The constant scales every density value and therefore scales the complete area. It must remain in the area equation before that equation is set equal to 1
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 12: Continuous probability distributions
In this chapter
12A Continuous random variables and density functions: Validate and normalise a continuous probability density function12A Continuous random variables and density functions: Estimate probabilities from grouped continuous frequency dataGroup numerical data into class intervalsConstruct a histogram from a frequency table12A Continuous random variables and density functions: Finite events, probability assignments and complements12A Continuous random variables and density functions: Conditional probability from exact representationsEvaluate and determine function rules12B Probability density validity and normalisation: Validate and normalise a continuous probability density function12B Probability density validity and normalisation: Estimate probabilities from grouped continuous frequency dataSolve a linear equation12B Probability density validity and normalisation: Evaluate a polynomial or recover a coefficient parameterWork with sets and interval notation12B Probability density validity and normalisation: Round to decimal places and significant figuresInterval and conditional probabilities from a densityCompound events, the addition rule and mutual exclusivity12C Interval and conditional probabilities: Finite events, probability assignments and complements12C Interval and conditional probabilities: Conditional probability from exact representationsMean, variance and standard deviation of a continuous random variableMedian, quartiles and percentiles of a continuous random variableExpand and factor polynomial expressions12D Mean, variance, standard deviation and percentiles: Evaluate a polynomial or recover a coefficient parameterMove between roots, factors, polynomial rules and equationsSolve a quadratic exactly with the quadratic formula12D Mean, variance, standard deviation and percentiles: Round to decimal places and significant figuresLinear transformations and totals of independent continuous random variables