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Question Number 192375 by Spillover last updated on 16/May/23

$$ShowthatE\left(Z\right)=0andVar\left(Z\right)=1where$$$$Zisthestandardnormalvariable$$

Answered by mehdee42 last updated on 16/May/23

$$Weknow\because Z=\frac{x-\mu }{\sigma }$$$$\mathrm{\&}E\left(kx\right)=kE\left(x\right)\mathrm{\&}E(x+k)=E\left(x\right)+k;k\in \mathbb{R}$$$$\mathrm{\&}Var\left(kx\right)=k^{2}Var\left(x\right)\mathrm{\&}Var(x+k)=Var\left(x\right)$$$$\Rightarrow E\left(Z\right)=E\left(\frac{x-\mu }{\sigma }\right)=\frac{1}{\sigma }E(x-\mu )=\frac{1}{\sigma }(E\left(x\right)-\mu )=0✓$$$$\mathrm{\&}Var\left(Z\right)=Var\left(\frac{x-\mu }{\sigma }\right)=\frac{1}{{\sigma }^2}Var(x-\mu )=\frac{1}{{\sigma }^2}Var\left(x\right)=1✓$$

Commented by Spillover last updated on 17/May/23

$$thanks$$

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