Probabilities from a continuous density
For a continuous variable, an interval probability is the area under its density. Split an integral when the density rule changes, and use the intersection over the conditioning event for a conditional probability. The conditional formula requires Pr(B) > 0.
\[\Pr(a<X<b)=\int_a^b f(x)\,dx,\quad\Pr(A\mid B)=\frac{\Pr(A\cap B)}{\Pr(B)}\]
- Integrate the density An interval probability is the area under f(x) across the event interval \(\displaystyle \Pr(a<X<b)=\int_a^b f(x)\,dx\)
- Split at a density junction If the rule changes inside the event interval, integrate each branch over its own part \(\displaystyle \int_0^3f(x)\,dx=\int_0^1f_1(x)\,dx+\int_1^3f_2(x)\,dx\)
- Subtract lower from upper For an antiderivative, evaluate the upper terminal first and subtract the lower terminal \(\displaystyle \int_1^3 2x\,dx=[x^2]_1^3=9-1=8\)
- Use the intersection for a condition A conditional probability divides the probability of both events by the probability of the conditioning event. This requires Pr(B) > 0. \(\displaystyle \Pr(A\mid B)=\frac{\Pr(A\cap B)}{\Pr(B)},\quad\Pr(B)>0\)
Worked example
A continuous random variable X has a constant density of one quarter from zero to four inclusive, and zero elsewhere. Let A contain values greater than one and less than three, and B contain values greater than zero and less than two. Find the probability of A given B
\[\Pr(A\mid B)=\frac{\Pr(1<X<2)}{\Pr(0<X<2)}\]
- \(\displaystyle A\cap B=\{1<X<2\}\)The intersection contains the values satisfying both interval conditions
- \(\displaystyle \Pr(A\cap B)=\int_1^2\frac{1}{4}\,dx=\frac{1}{4}\)Integrate the density over the intersection interval
- \(\displaystyle \Pr(B)=\int_0^2\frac{1}{4}\,dx=\frac{1}{2}\)Integrate the same density over the conditioning interval
- \(\displaystyle \Pr(A\mid B)=\frac{1/4}{1/2}=\frac{1}{2}\)Divide the intersection probability by the conditioning probability
Common mistakes
Using x times the density for a probability
✕\(\displaystyle \Pr(1<X<3)=\int_1^3x\left(\frac{1}{4}\right)\,dx=1\)✓\(\displaystyle \Pr(1<X<3)=\int_1^3\frac{1}{4}\,dx=\frac{1}{2}\)
For this constant density, integrate over values greater than one and less than three. Multiplying by the variable belongs to a mean calculationUsing one branch across a junction
✕\(\displaystyle \Pr(0<X<2)=\int_0^2f_1(x)\,dx\)✓\(\displaystyle \Pr(0<X<2)=\int_0^1f_1(x)\,dx+\int_1^2f_2(x)\,dx\)
When the density rule changes at 1, the integral must be split at that junctionUsing the whole support as a conditional denominator
✕\(\displaystyle \Pr(A\mid B)=\frac{\Pr(A\cap B)}{1}\)✓\(\displaystyle \Pr(A\mid B)=\frac{\Pr(A\cap B)}{\Pr(B)}\)
The denominator for a conditional probability is the probability of the conditioning event B. Use this formula only when Pr(B) > 0Difficulty levels
introCalculate a single-branch polynomial interval probability
standardSplit a piecewise interval or calculate a direct probability from an exponential density
challengeCalculate a conditional probability with two exact integrals or a circular density
examSelect the correct setup and calculate a conditional probability from a generated density
Questions about this topic
Why is x multiplied by the density for a mean but not for a probability?+
A probability is the area under f(x). A mean weights each value by x, so its integrand is x f(x)
When must an interval integral be split?+
Split it whenever a density rule changes inside the interval. Integrate each branch over the part where that rule applies, then add the areas
What does the condition B do in Pr(A given B)?+
Provided Pr(B) > 0, restrict attention to B. Integrate the density over A intersect B, then divide by the probability of B
Can I use a calculator?+
A calculator is allowed unless a question is marked no 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