The Normal and Lognormal Distributions John Norstad j-norstad@northwestern.edu http://www.norstad.org February, 999 Updated: November 3, 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.
DEFINITIONS AND SUMMARY OF THE PROPOSITIONS Definitions and Summary of the Propositions Proposition : Proposition : Proposition 3: e (x µ) /σ dx x e (x µ) /σ dx µ x e (x µ) /σ dx µ + σ Definition The normal distribution N[µ, σ ] is the probability distribution defined by the following density function: e (x µ) /σ Note that Proposition verifies that this is a valid density function (its integral from to is ). Definition The lognormal distribution LN[µ, σ ] is the distribution of e X where X is N[µ, σ ]. Proposition 4: If X is N[µ, σ ] then E(X) µ and Var(X) σ. Proposition 5: If Y is LN[µ, σ ] then E(Y ) e µ+ σ and Var(Y ) e µ+σ (e σ ). Proposition 6: If X is N[µ, σ ] then ax + b is N[aµ + b, a σ ]. Proposition 7: If X is N[µ, σ], Y is N[µ, σ], and X and Y are independent, then X + Y is N[µ + µ, σ + σ]. n Corollary : If X i are independent N[µ, σ ] for i... n then X i is N[nµ, nσ ]. Corollary : LN[nµ, nσ ]. Proposition 8: If Y i are independent LN[µ, σ ] for i... n then The probability density function of LN[µ, σ ] is: x e (log(x) µ) /σ i n Y i is i
PROOFS OF THE PROPOSITIONS Proofs of the Propositions Proposition e (x µ) /σ dx First assume that µ and σ. Let: a e x / dx Then: a e x / e y / dxdy e (x +y )/ dxdy Apply the polar transformation x r cos θ, y r sin θ, dxdy rdrdθ: a e r / rdrdθ [ e ] r / dθ [ ( )]dθ a >, and we just showed that a, so we must have a. For the general case, apply the transformation y x µ σ, dy dx σ : e (x µ) /σ dx e y / σdy e y / dy
PROOFS OF THE PROPOSITIONS 3 Proposition x e (x µ) /σ dx µ First assume that µ and σ. x e x / n dx lim xe x / dx n lim [ e ] n x / n lim / ) ( e n [( e / )] lim n For the general case, apply the transformation y x µ σ, dy dx σ : x e (x µ) /σ dx Proposition 3 µ (µ + σy) e y / σdy e y / dy + σ y e y / dy µ + σ (by Proposition ) µ x e (x µ) /σ dx µ + σ First assume that µ and σ. Integrate by parts using f x, f, g e x /, g xe x / : n x e x / dx n fg dx f(n)g(n) f()g() (e / ) (ne / ) n n f gdx e x / dx
PROOFS OF THE PROPOSITIONS 4 Then: n ne / n x e x / dx e x / dx e x / dx ne / lim lim n x e x / dx n [ n ] e x / dx ne / n e x / / dx lim n ne lim n ne / (by Proposition ) All that remains is to show that the last limit above is. We do this using L Hôpital s rule: lim / lim n ne n n e n / lim n ne n / For the general case, apply the transformation y x µ σ, dy dx σ : x e (x µ) /σ dx µ µσ (µ + σy) e y / σdy σ e y / dy + y e y / dy + y e y / dy µ + µσ + σ (by Propositions and ) µ + σ
PROOFS OF THE PROPOSITIONS 5 Proposition 4 If X is N[µ, σ ] then E(X) µ and Var(X) σ. E(X) x e (x µ) /σ dx µ (by Proposition ) Var(X) E(X ) E(X) (by Proposition in reference []) x e (x µ) /σ dx µ µ + σ µ (by Proposition 3) σ Proposition 5 If Y is LN[µ, σ ] then E(Y ) e µ+ σ Var(Y ) e µ+σ (e σ ). and Y e X where X is N[µ, σ ]. First assume that µ : E(Y ) E(e X ) e σ e xσ x σ dx e x e x /σ dx e (x σ ) +σ 4 σ dx e (x σ ) /σ dx e σ (by Proposition ) e σ E(Y ) E(e X ) e σ e 4xσ x σ dx e x e x /σ dx e (x σ ) +4σ 4 σ dx e (x σ ) /σ dx e σ (by Proposition ) e σ Var(Y ) E(Y ) E(Y ) (by Proposition in reference []) e σ (e σ ) e σ e σ e σ (e σ )
PROOFS OF THE PROPOSITIONS 6 For the general case, apply the transformation y x µ, dy dx: E(Y ) E(e X ) e µ e x e (x µ) /σ dx e µ+y e y /σ dy e y e y /σ dy (by the same reasoning as for the case µ above) e µ e σ e µ+ σ E(Y ) E(e X ) e µ e x e (x µ) /σ dx e (y+µ) e y /σ dy e y e y /σ dy (by the same reasoning as for the case µ above) e µ e σ e µ+σ Var(Y ) E(Y ) E(Y ) (by Proposition in reference []) e µ+σ (e µ+ σ ) e µ+σ e µ+σ e µ+σ (e σ ) Proposition 6 If X is N[µ, σ ] then ax + b is N[aµ + b, a σ ]. Prob(aX + b < k) Prob(X < (k b)/a) (k b)/a e (x µ) /σ dx (apply the transformation y ax + b, dy adx) k k e y b ( a µ) /σ a dy (aσ) e (y (aµ+b)) /a σ dy The last term above is the cumulative density function for N[aµ + b, a σ ], so we have our result.
PROOFS OF THE PROPOSITIONS 7 Proposition 7 If X is N[µ, σ ], Y is N[µ, σ ], and X and Y are independent, then X + Y is N[µ + µ, σ + σ ]. First assume that X is N[, ] and Y is N[, σ ]. Then: Prob(X + Y < k) e x / e u /σ dxdu x+u<k (apply the transformation u σy) e (x +y )/ σdxdy x+σy<k e (x +y )/ dxdy () x+σy<k At this point we temporarily make the assumption that k. Figure shows the area over which we are integrating. It is the half of the plane below and to the left of the line x + σy k. Note the following relationships: Figure : Area of Integration x r cos θ r k cos θ + σ sin θ y r sin θ r cos θ + σr sin θ x + σy k tan α k/(k/σ) σ α arctan σ
PROOFS OF THE PROPOSITIONS 8 We ll use the following technique to prove the result. First we ll convert the double integral above to polar coordinates. Then we ll rotate the result by α so that the graph above becomes the one shown in below. Then we ll show that the resulting integral is the same as the one for Prob(Z < k) where Z is N[, + σ ]. Figure : Area of Integration Rotated Note that r cos θ x k, and r k/cosθ. Apply the polar transformation x r cos θ, y r sin θ, dxdy rdrdθ to equation (): Prob(X + Y < k) e (x +y )/ dxdy x+σy<k re r / drdθ x+σy<k 3π/+α π/+α 3π/+α π/+α re r / drdθ + k cos θ+σ sin θ re r / drdθ () Note that we have made use of the assumption that k at this point to split the area over which we are integrating into two regions. In the first region, θ varies from π/ + α to 3π/ + α, and the vector at the origin with angle θ does not intersect the line x + σy, so r varies from to. In the second region, θ
PROOFS OF THE PROPOSITIONS 9 varies from π/ + α to π/ + α, and the vector does intersect the line, so r varies from to k/(cos θ + σ sin θ). We want to apply the transformation λ θ α to rotate. We first must calculate what happens to the upper limit of integration in the last integral above under this transformation. k cos θ + σ sin θ We now apply some trigonometric identities: k cos(λ + α) + σ sin(λ + α) k cos(λ + arctan σ) + σ sin(λ + arctan σ) cos(λ + arctan σ) cos λ cos(arctan σ) sin λ sin(arctan σ) sin(λ + arctan σ) sin λ cos(arctan σ) + cos λ sin(arctan σ) cos(arctan σ) + σ sin(arctan σ) σ + σ (3) cos(λ + arctan σ) cos λ σ sin λ + σ (4) σ sin(λ + arctan σ) σ sin λ + σ cos λ + σ (5) Adding equations (4) and (5) gives: cos(λ + arctan σ) + σ sin(λ + arctan σ) cos λ + σ cos λ + σ cos λ( + σ ) + σ cos λ + σ (6) We can now do our rotation under the transformation λ θ α. Equations (), (3) and (6) give: Prob(X + Y < k) 3π/ π/ π/ π/ 3π/ π/ π/ π/ re r / drdλ + k cos λ +σ re r / drdλ [ e ] r / dλ + [ e r / ] k cos λ +σ dλ
PROOFS OF THE PROPOSITIONS 3π/ π/ π/ [] dλ + π/ k π/ π/ [ ] k e cos λ(+σ ) ( ) dλ e cos λ(+σ ) dλ (7) Now we turn our attention to evaluating Prob(Z < k) where Z is N[, + σ ]. Let W be another random variable which is also N[, +σ ]. We use a sequence of steps similar to the one above, only without the rotation: Prob(Z < k) Prob(Z < k, < W < ) /(+σ ) /(+σ ) z<k + σ e z + σ e w dzdw ( + σ e (z +w )/(+σ ) dzdw ) z<k ( + σ re r /(+σ ) drdθ ) ( + σ ) ( + σ ) ( + σ ) ( + σ ) z<k 3π/ π/ π/ k/cosθ π/ 3π/ π/ π/ π/ π/ k π/ re r /(+σ ) drdθ + re r /(+σ ) drdθ [ ( ] + σ )e r /(+σ ) dθ + [ ( ] + σ )e r /(+σ k/ cos θ ) dθ e cos θ(+σ ) dθ (8) Equations (7) and (8) are the same, so at this point we have completed our proof that Prob(X + Y < k) Prob(Z < k) when k. Now suppose that k <. Propositon 6 implies that for any normal random variable X with mean, X is also normally distributed with mean and the same variance as X. Let A X, B Y, and W Z. Then A is N[, ], B is N[, σ ], and W is N[, + σ ]. So we have: Prob(X + Y < k) Prob( (X + Y ) > k) Prob(A + B > k) Prob(A + B < k) Prob(W < k) (because k > ) Prob(W > k) Prob( Z > k) Prob(Z < k)
PROOFS OF THE PROPOSITIONS At this point we have shown that Prob(X + Y < k) Prob(Z < k) for all k. Thus the random variables X + Y and Z have the same cumulative density function. Z is N[, + σ ], so X + Y is also N[, + σ ]. This completes our proof for the case that X is N[, ] and Y is N[, σ ]. For the general case where X is N[µ, σ ] and Y is N[µ, σ ], let A (X µ )/σ and B (Y µ )/σ. By Property 6, A is N[, ] and B is N[, σ /σ ]. Thus: Prob(X + Y < k) Prob(A + B < (k µ µ )/σ ) (k µ µ )/σ + σ /σ e x /(+σ /σ ) dx (apply the transformation y σ x + µ + µ ) k σ + σ e (y (µ+µ)) /(σ +σ ) dy This is the cumulative density function for N[µ + µ, σ + σ ], so we have our full result. Corollary If X i N[nµ, nσ ]. are independent N[µ, σ ] for i... n then n i X i is This corollary follows immediately from Proposition 7. Corollary If Y i LN[nµ, nσ ]. are independent LN[µ, σ ] for i... n then n i Y i is This corollary follows immediately from Definition and Proposition 7.
PROOFS OF THE PROPOSITIONS Proposition 8 The probability density function of LN[µ, σ ] is: x e (log(x) µ) /σ Suppose X is LN[µ, σ ]. Thev X e Y where Y is N[µ, σ ]. Then: Prob(X < k) Prob(e Y < k) Prob(Y < log(k)) log(k) e (y µ) /σ dy (apply the transformation x e y, y log(x), dy x dx) k x e (log(x) µ) /σ dx
REFERENCES 3 References [] John Norstad. Probability review. http://www.norstad.org/finance, Sep.