The Normal and Lognormal Distributions

The Normal and Lognormal
Distributions
John Norstad [e-mail] http://www.norstad.org

February 2, 1999
Updated: November 3, 2011

Abstract
The basic properties of the normal and lognormal distributions, with full proofs.
We assume familiarity with elementary probability theory and with college-level calculus. 1

1

DEFINITIONS AND SUMMARY OF THE PROPOSITIONS

1

Definitions and Summary of the Propositions
∞

√

Proposition 1:
−∞

2
2
1 e−(x−µ) /2σ dx = 1
2πσ

∞

x√

Proposition 2:
−∞

2
2
1 e−(x−µ) /2σ dx = µ
2πσ

∞

x2 √

Proposition 3:
−∞

2
2
1 e−(x−µ) /2σ dx = µ2 + σ 2
2πσ

Definition 1 The normal distribution N [µ, σ 2 ] is the probability distribution defined by the following density function:
√

2
2
1 e−(x−µ) /2σ
2πσ

Note that Proposition 1 verifies that this is a valid density function (its integral from −∞ to ∞ is 1).
Definition 2 The lognormal distribution LN [µ, σ 2 ] is the distribution of eX where X is N [µ, σ 2 ].
Proposition 4: If X is N [µ, σ 2 ] then E(X) = µ and Var(X) = σ 2 .
1

2

Proposition 5: If Y is LN [µ, σ 2 ] then E(Y ) = eµ+ 2 σ and
2
2
Var(Y ) = e2µ+σ (eσ − 1).
Proposition 6: If X is N [µ, σ 2 ] then aX + b is N [aµ + b, a2 σ 2 ].
2
2
Proposition 7: If X is N [µ1 , σ1 ], Y is N [µ2 , σ2 ], and X and Y are indepen2
2
dent, then X + Y is N [µ1 + µ2 , σ1 + σ2 ]. n Corollary 1:

2

If Xi are independent N [µ, σ ] for i = 1 . . . n then

Xi is i=1 N [nµ, nσ 2 ]. n Corollary 2:

2

If Yi are independent LN [µ, σ ] for i = 1 . . . n then

Yi is i=1 LN [nµ, nσ 2 ].
Proposition 8: The probability density function of LN [µ, σ 2 ] is:
2
2
1
√ e−(log(x)−µ) /2σ x 2πσ

2

2

PROOFS OF THE PROPOSITIONS

2

Proofs of the Propositions

Proposition 1
∞

√
−∞

2
2
1 e−(x−µ) /2σ dx = 1
2πσ

Proof:
First assume that µ = 0 and σ = 1. Let:
∞

a=
−∞

2
1
√ e−x /2 dx
2π

Then:
∞

a2