Show-that-E-Z-0-and-Var-Z-1-where-Z-is-the-standard-normal-variable-

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

S h o w t h a t E (Z) = 0 a n d V a r (Z) = 1 w h e r e

Z i s t h e s t a n d a r d n o r m a l v a r i a b l e

Answered by mehdee42 last updated on 16/May/23

W e k n o w ∵ Z = \frac{x - μ}{σ}

& E (k x) = k E (x) & E (x + k) = E (x) + k; k \in R

& V a r (k x) = k^{2} V a r (x) & V a r (x + k) = V a r (x)

\Rightarrow E (Z) = E (\frac{x - μ}{σ}) = \frac{1}{σ} E (x - μ) = \frac{1}{σ} (E (x) - μ) = 0 ✓

& V a r (Z) = V a r (\frac{x - μ}{σ}) = \frac{1}{σ^{2}} V a r (x - μ) = \frac{1}{σ^{2}} V a r (x) = 1 ✓

Commented by Spillover last updated on 17/May/23

t h a n k s