Page 204 - Applied statistics and probability for engineers

182 Chapter 5/Joint Probability Distributions

The following results can be shown for a bivariate normal distribution. The details are left
as an exercise.
Marginal
Distributions of If X and Y have a bivariate normal distribution with joint probability density f xy ( x y,σ X ,
,
Bivariate Normal
,
,
,
Y
X
Y
Random Variables σ μ μ ρ), the marginal probability distributions of X and Y are normal with means
μ X and μ y and standard deviations σ X and σ y , respectively. (5-21)
Conditional

Distribution of If X and Y have a bivariate normal distribution with joint probability density f XY ( x, y,
Bivariate Normal σ σ Y , μ X , μ Y , ρ), the conditional probability distribution of Y given X = x is normal
X ,
Random Variables with mean σ
= μ + ρ Y
X
μ |Y x Y σ x ( − μ )
X
and variance
2 2 − ρ )
2
σ |Y x = σ ( 1
Y
Furthermore, as the notation suggests, r represents the correlation between X and Y. The fol-
lowing result is left as an exercise.
Correlation of
Bivariate Normal If X and Y have a bivariate normal distribution with joint probability density function
Random Variables f XY ( x y,σ x ,σ y ,μ x ,μ y , )ρ , the correlation between X and Y is r. (5-22)
,

The contour plots in Fig. 5-16 illustrate that as r moves from 0 (left graph) to 0.9 (right graph),
the ellipses narrow around the major axis. The probability is more concentrated about a line in
=
x
y
the ( , ) plane and graphically displays greater correlation between the variables. If ρ −1 or
(
+1, all the probability is concentrated on a line in the x,y) plane. That is, the probability that
Xand Y assume a value that is not on the line is zero. In this case, the bivariate normal prob-
ability density is not deined.
In general, zero correlation does not imply independence. But in the special case that X
and Y have a bivariate normal distribution, if p = 0, X and Y are independent. The details are
left as an exercise.
For Bivariate Normal
Random Variables If X and Y have a bivariate normal distribution with r = 0, X and Y
Zero Correlation are independent. (5-23)
Implies Independence

An important use of the bivariate normal distribution is to calculate probabilities involving
two correlated normal random variables.

f XY (x, y) y

FIGURE 5-17
Bivariate normal y 0
probability density 0
function with σ x = 1,
σ y = 1, ρ = 0, μ x = 0 0 x
μX = 0, and μ y = 0. 0 x