Applied Statistics Using SPSS, STATISTICA, MATLAB and R

424 Appendix A - Short Survey on Probability Theory
A: Note first that the domain where the density is non-null corresponds to a
triangle of area ½. Therefore, the total volume under the density function is 1 as it
should be. The marginal distributions and densities are computed as follows:
x1
∞
x1
1
2
F1
( x1)
= f ( u,
v)
dudv ⎜
⎛ 2dv⎟
⎞
∫ = du = 2x
0
1 − x
−∞
∫−∞
∫ u
1
⎝∫
⎠
dF1
( x1)
⇒ f1
( x1)
= = 2 − 2x1
dx
∞ x2
x2v 2
dF2
( x2
)
F2
( x2
) = f ( u,
v)
dudv ⎜
⎛ 2du⎟
⎞
∫ =
dv = x2
⇒ f 2 ( x2
) = = 2x
2.
−∞∫−∞ ∫0 ⎝∫0
⎠
dx
The probability is computed as:
P(
X
1
≤ ½, X 2 ≤ ½) = ∫ ∫ 2dudv
= ∫
½
v
−∞−∞ ½
0
2vdv
= ¼ .
The same result could be more simply obtained by noticing that the domain has
an area of 1/8.
f(x,y)
x y
Figure A.9. Bell-shaped surface of the bivariate normal density function.
The bivariate normal density function has a bell-shaped surface as shown in
Figure A.9. The equidensity curves in this surface are circles or ellipses (an
example of which is also shown in Figure A.9). The probability of the event
(x1 ≤ X < x2, y1 ≤ Y < y2) is computed as the volume under the surface in the
mentioned interval of values for the random variables X and Y.
The equidensity surfaces of a trivariate normal density function are spheres or
ellipsoids, and in general, the equidensity hypersurfaces of a d-variate normal
density function are hyperspheres or hyperellipsoids in the d-dimensional
space, ℜ d .
1
2