The Normal and Lognormal Distributions
John Norstad j-norstad@northwestern.edu 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.

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DEFINITIONS AND SUMMARY OF THE PROPOSITIONS

1

1

Definitions and Summary of the Propositions
∞

Proposition 1:
−∞ ∞

√

2 2 1 e−(x−µ) /2σ dx = 1 2πσ 2 2 1 e−(x−µ) /2σ dx = µ 2πσ 2 2 1 e−(x−µ) /2σ dx = µ2 + σ 2 2πσ

Proposition 2:
−∞ ∞

x√

Proposition 3:
−∞

x2 √

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 . 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
1 2

Corollary 1: N [nµ, nσ 2 ].

If Xi are independent N [µ, σ ] for i = 1 . . . n then i=1 2

Xi is

n

Corollary 2: LN [nµ, nσ 2 ].

If Yi are independent LN [µ, σ ] for i = 1 . . . n then i=1 2

Yi is

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

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PROOFS OF THE PROPOSITIONS

2

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