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Linear transformations and totals of independent continuous random variables

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Transformation
\(\displaystyle Y=-2X+5\)

Which rule gives the variance of the transformed variable?

Year 12, VCE Units 3 & 4, Chapter 12, 4 difficulty levels, Calculator allowedBuild a worksheet

Linear transformations of moments

A linear transformation changes the centre by multiplying and adding. It changes spread through the multiplier only, while independent totals add their means and variances

\[E(aX+b)=aE(X)+b,\quad\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X),\quad\operatorname{SD}(aX+b)=|a|\operatorname{SD}(X)\]
  1. Transform the mean linearly Multiply the mean by the coefficient of X, then add the constant \(\displaystyle E(aX+b)=aE(X)+b\)
  2. Square the coefficient for variance The constant changes location but not spread, so it is absent from the variance rule \(\displaystyle \operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)\)
  3. Use the absolute coefficient for standard deviation Standard deviation is non-negative, so a negative coefficient is changed to its absolute value \(\displaystyle \operatorname{SD}(aX+b)=|a|\operatorname{SD}(X)\)
  4. Add moments for independent totals For independent identically distributed variables, means and variances each add across the copies \(\displaystyle E(S_n)=nE(X),\quad\operatorname{Var}(S_n)=n\operatorname{Var}(X)\)
  5. Recover the second moment Add the variance to the square of the mean \(\displaystyle E(X^2)=\operatorname{Var}(X)+[E(X)]^2\)

Worked example

Use the displayed mean and variance of the continuous random variable to find its second moment and the mean, variance and standard deviation of its linear transformation. For a second example, take nine independent, identically distributed copies of the transformed variable and find the mean, variance and standard deviation of their total

\[\begin{gathered}E(X)=4,\quad\operatorname{Var}(X)=\frac94\\Y=-2X+7\\S=\sum_{i=1}^{9}Y_i\end{gathered}\]
  1. \(\displaystyle E(Y)=(-2)(4)+7=-1\)Multiply the mean by -2 and then add 7.
  2. \(\displaystyle \operatorname{Var}(Y)=(-2)^2\left(\frac{9}{4}\right)=9\)Square the multiplier and leave the added constant out of the variance
  3. \(\displaystyle \operatorname{SD}(X)=\sqrt{\frac{9}{4}}=\frac{3}{2}\)Take the principal square root of the original variance
  4. \(\displaystyle \operatorname{SD}(Y)=|-2|\left(\frac{3}{2}\right)=3\)Multiply the original standard deviation by the absolute value of the coefficient
  5. \(\displaystyle E(X^2)=\operatorname{Var}(X)+[E(X)]^2\)Recover the second moment of the original variable by adding its variance to its squared mean
  6. \(\displaystyle E(X^2)=\frac94+4^2=\frac94+16=\frac{73}{4}\)Substitute the displayed original moments and use a common denominator
  7. \(\displaystyle E(S)=\sum_{i=1}^{9}E(Y_i)=9E(Y)\)Linearity adds the means and identical distributions give the same mean for every copy
  8. \(\displaystyle E(S)=9\times(-1)=-9\)Multiply the transformed mean by the number of copies
  9. \(\displaystyle \operatorname{Var}(S)=\sum_{i=1}^{9}\operatorname{Var}(Y_i)=9\operatorname{Var}(Y)\)Independence allows the variances to add and identical distributions give the same variance for every copy
  10. \(\displaystyle \operatorname{Var}(S)=9\times(-2)^2\times\frac94=81\)Multiply the transformed variance by the number of independent copies, not its square
  11. \(\displaystyle \operatorname{SD}(S)=\sqrt{\operatorname{Var}(S)}=\sqrt{81}=9\)Take the non-negative principal square root after adding the variances

Common mistakes

Not squaring the multiplier
✕\(\displaystyle \operatorname{Var}(-2X+5)=-2\left(\frac{9}{4}\right)=-\frac{9}{2}\)✓\(\displaystyle \operatorname{Var}(-2X+5)=(-2)^2\left(\frac{9}{4}\right)=9\)
Variance uses the square of the multiplier, so a negative multiplier cannot make the variance negative
Multiplying the added constant in the mean
✕\(\displaystyle E(3X+4)=3E(X)+3(4)\)✓\(\displaystyle E(3X+4)=3E(X)+4\)
The constant is added after the mean is multiplied; it is not multiplied by the coefficient of X.
Adding standard deviations for a total
✕\(\displaystyle \operatorname{SD}(S_4)=4\operatorname{SD}(X)\)✓\(\displaystyle \operatorname{SD}(S_4)=\sqrt{4\operatorname{Var}(X)}=2\operatorname{SD}(X)\)
For independent copies, variances add first and the square root is taken once

Difficulty levels

introFind a transformed mean or variance from given moments
standardRecover a second moment and find a transformed standard deviation with a negative multiplier
challengeFind moments from a stated density before applying a linear transformation
examFind a total mean, variance or standard deviation for independent copies

Questions about this topic

Why does the added constant not affect variance?+

Adding a constant moves every value by the same amount, so each distance from the mean is unchanged. The spread therefore depends on the multiplier only

Why is the absolute value used for standard deviation?+

Standard deviation cannot be negative. A negative multiplier reverses the order of values but changes their spread by its positive magnitude

How do moments behave for a total of independent copies?+

The means add and the variances add. The standard deviation is the principal square root of the total variance, not the sum of individual standard deviations

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