Download Theorem If X ∼ N(0,1), then Y = µ + σX has the normal distribution

Theorem If X ∼ N (0, 1), then Y = µ + σX has the normal distribution with mean µ and
variance σ 2 .
Proof Let the random variable X ∼ N (0, 1). The probability density function of X is
2
e−x /2
fX (x) = √
2π
− ∞ < x < ∞.
The transformation Y = µ + σX is a 1–1 transformation from X = {x | − ∞ < x < ∞} to
Y = {y | − ∞ < y < ∞}, with inverse X = (Y − µ)/σ and Jacobian
dX
1
= .
dY
σ
Therefore, by the transformation technique, the probability density function of Y is
fY (y) =
=
dx fX (g −1 (y)) dy
−(y−µ)2 /2σ 2 e
1
√
σ
2π
(y−µ)2
1
e− 2σ2
= √
2πσ
− ∞ < x < ∞,
which is the probability density function of a N (µ, σ 2 ) random variable.
APPL verification: The APPL statements
X := StandardNormalRV():
assume(sigma > 0);
g := [[x -> mu + sigma * x], [-infinity,infinity]]:
Y := Transform(X, g);
yield the probability density function of a N (µ, σ 2 ) random variable.
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