Search topics

Search

Mean, variance and standard deviation of a continuous random variable

Start a set of 8
Step 3 of 5

Compute the expected value for the displayed density

\[\int_0^3x\left(\frac{2x}{9}\right)\,dx=\frac{2}{9}\left[\frac{x^3}{3}\right]_0^3\]
Year 12, VCE Units 3 & 4, Chapter 12, 4 difficulty levels, Calculator allowedBuild a worksheet

Centre and spread from a density

A continuous density gives the mean through a value-weighted integral. The second moment then gives variance, and the square root of variance gives standard deviation

\[E(X)=\int_a^b xf(x)\,dx,\quad \operatorname{Var}(X)=E(X^2)-[E(X)]^2\]
  1. Weight values by the density Multiply each value by its density before integrating over the support \(\displaystyle E(X)=\int_a^b xf(x)\,dx\)
  2. Find the second moment separately Use squared values to calculate the second moment before finding variance \(\displaystyle E(X^2)=\int_a^b x^2f(x)\,dx\)
  3. Subtract the squared mean Variance is the second moment minus the square of the mean \(\displaystyle \operatorname{Var}(X)=E(X^2)-[E(X)]^2\)
  4. Take the principal square root Standard deviation is the non-negative square root of variance \(\displaystyle \operatorname{SD}(X)=\sqrt{\operatorname{Var}(X)}\)

Worked example

A continuous random variable has the density shown below and is zero elsewhere. Find its mean and variance

\[f(x)=\frac{2x}{9},\quad 0\le x\le 3\]
  1. \(\displaystyle E(X)=\int_0^3x\left(\frac{2x}{9}\right)\,dx=\frac{2}{9}\int_0^3x^2\,dx\)Use x times the density to form the expected-value integral
  2. \(\displaystyle E(X)=\frac{2}{9}\left[\frac{x^3}{3}\right]_0^3=\frac{2}{9}(9-0)=2\)Evaluate the integral over the complete support to obtain the mean
  3. \(\displaystyle E(X^2)=\int_0^3x^2\left(\frac{2x}{9}\right)\,dx=\frac{2}{9}\left[\frac{x^4}{4}\right]_0^3=\frac{2}{9}\left(\frac{81}{4}-0\right)=\frac{9}{2}\)Use x squared times the density to obtain the second moment
  4. \(\displaystyle \operatorname{Var}(X)=\frac{9}{2}-2^2=\frac{1}{2}\)Subtract the squared mean from the second moment

Common mistakes

Using the density alone for the mean
✕\(\displaystyle E(X)=\int_0^3f(x)\,dx=1\)✓\(\displaystyle E(X)=\int_0^3xf(x)\,dx=2\)
Integrating f(x) alone gives the total area, not the value-weighted mean
Using the squared mean as the second moment
✕\(\displaystyle E(X^2)=[E(X)]^2=2^2=4\)✓\(\displaystyle E(X^2)=\int_0^3x^2f(x)\,dx=\frac{9}{2}\)
The second moment must be found by integrating x squared times the density; it is not usually the squared mean
Reporting variance as standard deviation
✕\(\displaystyle \operatorname{SD}(X)=\operatorname{Var}(X)=\frac{1}{2}\)✓\(\displaystyle \operatorname{SD}(X)=\sqrt{\operatorname{Var}(X)}=\frac{\sqrt{2}}{2}\)
Standard deviation is the principal square root of variance, not variance itself

Difficulty levels

introFind the expected value for a simple linear density on a bounded interval
standardFind variance or exact standard deviation for a polynomial or piecewise-linear density
challengeFind a stated-precision standard deviation for a less simple density
examSelect a moment integral and compute a consequence from a generated density

Questions about this topic

Why is the mean not necessarily the value with the greatest density?+

The mean is a weighted average of all values in the support. The highest density identifies where values are most concentrated, not the average value

What is the difference between variance and standard deviation?+

Variance is the average squared distance from the mean. Standard deviation is the principal square root of variance and has the same units as X.

Why is the second moment calculated separately?+

Variance uses the second moment and the square of the mean, which are generally different. Calculate the second moment with its own integral

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