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Distribution of a linear transformation of a discrete random variable

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One draw, transformed values

For Y=aX+b, take one value of X and apply the deterministic rule. If a≠0, distinct X values remain distinct, so each value keeps its paired probability. Sort the new values and probabilities together, especially when a<0. Never multiply a probability by a or add b to it

Year 11, VCE Units 1 & 2, Chapter 6, 4 difficulty levels, No calculatorBuild a worksheet

Scaling and shifting one variable

Transform values; preserve or combine their probability mass

  1. Mean E(aX+b)=aμ+b.
  2. Variance Var(aX+b)=a²σ²
  3. SD SD(aX+b)=|a|σ
  4. Degenerate case If a=0, all mass is at b.

Worked example

Find the distribution, mean and variance of Y from the following data

\[\begin{aligned}\Pr(X=-1)&=\frac23,\quad\Pr(X=2)=\frac13\\Y&=-2X+3\\E(X)&=0,\quad E(X^2)=2\end{aligned}\]
  1. \(\displaystyle \begin{aligned}Y|_{X=-1}&=-2(-1)+3=5, &\Pr(Y=5)&=\frac23\\Y|_{X=2}&=-2(2)+3=-1, &\Pr(Y=-1)&=\frac13\end{aligned}\)Map each original row and then sort the transformed value-probability pairs
  2. \(\displaystyle \begin{aligned}E(Y)&=-\frac13+\frac{10}3=3\\E(Y^2)&=\frac13+\frac{50}3=17\\\operatorname{Var}(Y)&=17-3^2=8\end{aligned}\)Check the mean and second moment directly, then subtract the square of the mean

Common mistakes

Changing probabilities
✕Pr(Y=ax+b)=a Pr(X=x)+b✓For a≠0, keep the probability paired with x
The deterministic rule changes values; probability mass is reassigned, not scaled
Signed SD
✕SD(−2X)=−2 SD(X)✓SD(−2X)=2 SD(X)
SD is the nonnegative square root of variance

Difficulty levels

introMap values and use exact moment rules
standardComplete transformed distributions and moments
challengeNegative transformations and event comparisons
examCombined finite distributions, moments and events

Questions about this topic

Why do practice questions use nonzero a and b?+

That is the frozen bank’s parameter scope. The lesson explicitly teaches and checks zero multiplier and zero shift cases

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 6: Random variables and sampling

In this chapter

Expected value and linear expectation of a discrete random variableVariance, standard deviation and linear variance of a discrete random variableDistribution of a linear transformation of a discrete random variableConstruct finite independent sums and compare repeated copies with scalingDesign and interpret a probability simulationPopulations, samples, parameters, statistics and sampling methodsExact sampling distribution of a sample proportionConstruct sample means and interpret finite sampling distributions