Variance of fixed minus non-fixed points

Question

Shuffle the numbers $1$ through $1000$ uniformly at random. Let $Y$ be the number of fixed points and $X$ the number of points that are moved. Find $\operatorname{Var}(Y-X)$ .

Short answer

$4$ . Since $X=1000-Y$ , $Y-X=2Y-1000$ and the variance is $4\operatorname{Var}(Y)$ . With indicators, $E[Y]=1$ and $E[Y(Y-1)]=1000\cdot999\cdot\tfrac{1}{1000\cdot999}=1$ , so $\operatorname{Var}(Y)=1+1-1=1$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Counting and Combinatorics
Conditional Probability, Total Probability, and Bayes' Rule

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