5.12 The Bivariate Normal Distribution 313 512 The Bivariate Normal Distribution

The first multivariate continuous distribution for which we have a name is a generalization of the normal distribution to two coordinates. There is more structure to the bivanate normal distribution than just a pair of normal marginal distributions.

Definition of the Bivarlate Normal Distribution

Suppose that Z and Z are independent random variables, each of which has a standard normal distribution. Then the joint p.d.f. 2) of 1Z and Z is specified for all values of and z by the equation

g(z,z,) = — exp[——(z + z)]. (5.12.1) 1 2 L 2 For constants 1t. /12, a. a,, and p such that —00 0 (i = 1, 2). and —1< p < 1, we shall now define two new random variables 1X and X, as follows: 1X 1=aZ 1-i—i X, = a [pZi + (1 — p 2z + it,. (5.12.2) We shall derive the joint p.d.f. 1f(x X2) of 1X and X,. The transformation from 1Z . and 1, to 1X and 2X is a linear transformation; and it will be found that the determinant of the matrix of coefficients of 1Z and 2Z has the value z\ = (1 Therefore, as discussed — )p2 a1212 in Section 3.9, the Jacobian J of the inverse transformation. from 1X and K, to 1Z and 2Z is J = —i = -aa2(l—p-) . (5.12.3) Since J > 0. the value of IJI is equal to1the value of J itself. If the relations (5.12.2) are solved for Zi and 2Z in terms of X and .2X then the joint p.d.f. 1f(x x-,) can be obtained by replacing and z in Eq. (5.12.1) by their expressions in terms. 1ofx and 52. and then multiplying by JI. It can be shown that the result is, for — 1

When thejoint p.d.f. of two random variables 1X and 2X is of the form in Eq. (5.12.4). it is said that 1X and X-,have a biiari ate normaldistribution. The means and the variances of the bivariate normal distribution specified by Eq. (5.12.4) are easily derived from the definitions in Eq. (5.12.2). Because 1Z and 2Z are independent and each has mean 0 and

314

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5.12.1

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Tao random variables 1.V and X that hake a baariate normal distribution are independent if and only if they are uncorrelated.

\Ve have already seen in Section 4.6 that tso random sariables X and X v ith an arbitrary joint distribution can he unconelated ithout being independent.

(‘onditional Distributions. The conditional distribution of X-. izien that ‘i = can also he derived from the representation in Eq. 15.12.2). If 1X = .. . then 1Z = )• Therefore, the conditional I — distribution of ,V eien that .V = i is the same as the conditional distribution of 2)t (I 2J7 + /L + (i (5.12.5) at I Because Z has a standard normal distribution and is independent of .1V it follows from 5. 12.5 that the conditional distribution of .V cisen that X = .r is a normal distribution. for which the mean is . _/tt), E(Xx = p. + pa (5.12.6) )1 ( and the variance is (1 — p2)a. The conditional distribution of 1X given that X = x cannot he deri’ed so easily from Eq. 5. 12.2) because of the different ways in s hich Z and Z enter Eq. t 5. 12.2). f1oseer. it is seen from Eq. (5.12.4) that the Joint p.d.f. 1fLy v is symmetric in the I t o ariables — p I o and (v /L ) nH . Therefore, it. foIlos s ihat the conditional distribution of 1X gi en that X = r can he found from the conditional distribution of \ uRen that .V = Ithis distribLttion has Just been deri’.ed) simply by interchanging

Example

316

Example

5.12.3

5,12,2

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If is 5.1.2 The Bivariate Normal Distribution 317

.Vand Y must he a hi ariate nirmal distribution see Exercise 14>.Hence, the marginal

distribution of 1 is also a normal disinbution. [Themean and the ariance of 1 niust he determined.

ihe mean at } is EiY EIEiYXiJ EX —jr.

Eui themmore. h E\ercmse II of Section 4.7,

ar( Y> E[Var( Y X ii Var[ Ei Y Xi Er) —Var>X>

k Hence, he distribution of ) is a normal distribution s oh mean i and variance r- o .4

Linear Combinations Suppose again that two random variables X and X hake a hivariate normal distribution, tar shich the p.d.f. is specified by Eq. 5.12.4). Now consider the random variable Y = a X + a X + /, v here 0. a. and h are arbitrary given constants. Both 1X and X can he represented. as in Eq. (5.12.2). as linear combinations of independent and normally distributed random variables Z and Z. Since Y is a linear combination of X and X. it follows that V can also be represented as a linear combination of 1Z and Z. Therefore. by Corollary 56.1. the distribution of V will also be a normal distribution. Thus, the following important property has been established.

If two random variables 1X and ,V have a b/variate normal distribution, then will distribution. each linear combination V = a1X -i-aX + h have a normal

The mean and variance of V are as follows: E>Y> = a1E(X + a2E(X + b ) +a2M2 +1 ‘ili ) and Van Y = 02 Var) )1K a )5VariX a12a 1Cov>X )2K = c1rT aa. ± 2aapon. , (5.12.8) Example 5.12.4 Heights of Husbands and Wives. Suppose that a maiTied couple is selected at random from a certain population of married couples. and that the joint distribution of the height of the s ife and the height of her husband is a bivariate normal distribution. Suppose that the heights at the wives have a mean of 66.8 inches and a standard deviation of 2 inches. the heights of the husbands have a mean of 70 inches and a standard deviation of 2 inches. and the correlation heteen these to heights is 0.68. We shall determine the probability that the wife will he taller than her husband.

EXERCISES

2.

1.

318

score

population.

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given

Suppose

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15

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