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Generating Correlated Random Variables Bivariate Gaussian Distribution The joint (bivariate) PDF for X 1,2 is f X 1 X 2 (x 1 ,x 2 )= 1 2π 1 (1 - ρ 2 ) 1/2 exp - 1 2(1 - ρ 2 ) x 2 1 σ 2 1 + x 2 2 σ 2 2 - 2ρ x 1 x 2 σ 1 σ 2 A more useful of writing this PDF is to use the column vector X = col (X 1 ,X 2 ) and the covariance matrix C = σ 2 1 σ 1 σ 2 ρ σ 1 σ 2 ρ σ 2 2 to write (using to denote transpose) f X (X)= 1 2π (det C) 1/2 exp - 1 2 X C -1 X . The bivariate Gaussian is used frequently in likelihood and Bayesian estimation to display contours for parameter estimates. 1

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Page 1: Generating Correlated Random Variables - Cornell Universityhosting.astro.cornell.edu/.../GeneratingCorrelatedRandomVariables.pdf · Generating Correlated Random Variables Bivariate

Generating Correlated Random Variables

Bivariate Gaussian Distribution

The joint (bivariate) PDF for X1,2 is

fX1X2(x1, x2) =1

1

(1− ρ2)1/2exp

− 1

2(1− ρ2)

x21σ21

+x22σ22− 2ρ

x1x2σ1σ2

A more useful of writing this PDF is to use the column vector X = col (X1, X2) and thecovariance matrix

C =

σ21 σ1σ2ρ

σ1σ2ρ σ22

to write (using † to denote transpose)

fX(X) =1

2π(detC)1/2exp

−12X†C−1X

.

The bivariate Gaussian is used frequently in likelihood and Bayesian estimation to displaycontours for parameter estimates.

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Page 2: Generating Correlated Random Variables - Cornell Universityhosting.astro.cornell.edu/.../GeneratingCorrelatedRandomVariables.pdf · Generating Correlated Random Variables Bivariate

Figure 1: Scatter plots of two random variables X1,2 that have a joint Gaussian PDF for four different values of correlation coefficient, ρ.

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Page 3: Generating Correlated Random Variables - Cornell Universityhosting.astro.cornell.edu/.../GeneratingCorrelatedRandomVariables.pdf · Generating Correlated Random Variables Bivariate

Generating Correlated Random Variables

Consider a (pseudo) random number generator that gives numbers consistent with a 1D Gaus-sian PDF ≡ N(0, σ2) (zero mean with variance σ2).

How do we create two Gaussian random variables (GRVs) from N(0, σ2) but that are correlatedwith correlation coefficient ρ?

So we wantρX1,X2 =

〈(X1 − 〈X1〉) (X2 − 〈X2〉)〉σ2

.

Define Y1, Y2 as independent N(0, σ2) GRVs, so ρY1,Y2 = 0 and let

X1 = aY1 + bY2

X2 = cY1 + dY2.

Since the means of all variables are zero, we have

〈X1X2〉 = 〈(aY1 + bY2)(cY1 + dY2)〉= ac〈Y 2

1 〉 + bd〈Y 22 〉 + (ad + bc)〈Y1Y2〉

= (ac + bd)σ2

ThereforeρX1X2 =

〈X1X2〉σ2

= ac + bd (1)

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Page 4: Generating Correlated Random Variables - Cornell Universityhosting.astro.cornell.edu/.../GeneratingCorrelatedRandomVariables.pdf · Generating Correlated Random Variables Bivariate

We also want〈X2

1〉 = (a2 + b2)σ2 = σ2

〈X22〉 = (c2 + d2)σ2 = σ2

so

a2 + b2 = c2 + d2 = 1 (2)

A natural solution is to use a = cosφ b = sinφ

c = sinφ d = cosφ.

Then the constraint equations (1) and (2) are satisfied and

ac + bd = ρX1X2 = 2 cosφ sinφ = sin 2φ

soφ =

1

2sin−1 ρX1X2.

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