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LearnEdexcel9FM0First teaching 2017Option · Further Statistics 2Combinations of random variables

Option · Module

Combinations of random variables

3 subtopics≈ 100 min in total

Start the module: E and Var of aX + b and aX + bY Open graphing calculator

E and Var of aX + b and aX + bY
Expectations behave exactly as you would hope: $E(aX+bY)=aE(X)+bE(Y)$ . Variances need care: constants are squared, added constants vanish, and for independent variables the variances always add, even for a difference.
≈ 30 min
Sums of observations and multiples
Weighing one bag and multiplying by $4$ is not the same as weighing four different bags. Both totals have the same mean, but $4X$ has variance $16\sigma^2$ while $X_1+X_2+X_3+X_4$ has only $4\sigma^2$ .
≈ 35 min
Combining normal and Poisson variables
Any linear combination of independent normals is normal, so "is $X$ bigger than $Y$ ?" becomes "is $X-Y$ positive?". Poisson variables add to a Poisson, but only sums: $2X$ and $X-Y$ are not Poisson.
≈ 35 min