Marginal and Conditional Probability Distributions

Marginal and Conditional Probability Distributions

With each two-dimensional random variable (X, Y) we associate two onedimensional
random variables, namely X and Y, individually. That is, we may be
interested in the probability distribution of X or the probability distribution of Y.

EXAMPLE 1
Let us again consider .Let us also compute the "marginal" totals, that is, the sum of the 6 columns and 4 rows of the table. (See Table 1)
          The probabilities appearing in the row and column margins represent the probability distribution of Y and X, respectively. For instance, P( Y = l) = 0.26, P(X = 3) = 0.21, etc. Because of the appearance of Table 1 we refer, quite generally, to the marginal distribution of X or the marginal distribution of Y, whenever we have a two-dimensional random variable (X, Y), whether discrete or continuous.

X\Y	0	1	2	3	4	5	Sum
0	0	0.01	0.03	0.05	0.07	0.09	0.25
1	0.01	0.02	0.04	0.05	0.06	0.08	0.26
2	0.01	0.03	0.05	0.05	0.05	0.06	0.25
3	0.01	0.02	0.04	0.06	0.06	0.05	0.24
Sum	0.03	0.08	0.16	0.21	0.24	0.28	1.00

In the discrete case we proceed as follows: Since X = x_i must occur with Y = y_j for some j and can occur with Y = y_j for only one j, we have
p(x_i) = P(X = x_i) = P(X = x_i, Y = y₁ or X = x_i, Y = y₂ or···)
\[\sum\limits_{j=1}^{\infty }{p\left( {{x}_{i}},{{y}_{j}} \right)}\] The function p defined for x1, x2, • • . , represents the marginal probability distribution of X Analogously we define q(y_j) = P( Y y_j) = \[\sum\limits_{j=1}^{\infty }{p\left( {{x}_{i}},{{y}_{j}} \right)}\]  as the marginal probability distribution of Y. In the continuous case we proceed as follows: Let f be the joint pdf of the continuous two-dimensional random variable ( X, Y). We define g and h, the marginal probability density functions of X and Y, respectively, as follows: \[g\left( x \right)=\int_{-\infty }^{+\infty }{f\left( x,y \right)}dy;\] \[h\left( y \right)=\int_{-\infty }^{+\infty }{f\left( x,y \right)}dx;\]

These pdrs correspond to the basic pdrs of the one-dimensional random variables X and Y, respectively.
For example
P( c ≤ X ≤  d) = P [ c ≤ X ≤  d, - ∞ < Y < ∞]

\[\int_{c}^{d}{\int_{-\infty }^{+\infty }{f\left( x,y \right)dydx=\int_{c}^{d}{g\left( x \right)dx}}}\]

EXAMPLE 2
Two characteristics of a rocket engine's performance are thrust X and mixture ratio Y. Suppose that (X, Y) is a two-dimensional continuous random variable with joint pdf:

f(x, y) = 2(x + y - 2xy),              0 ≤ x ≤  1, 0 ≤  y ≤  1,
           = 0,                                    elsewhere.

(The units have been adjusted in order to use values between 0 and 1.) The marginal
pdf of X is given by

\[g\left( x \right)=\int_{0}^{1}{2\left( x+y-2xy \right)}dy=2\left( xy+\frac{{{y}^{2}}}{2}-x{{y}^{2}} \right)|_{0}^{1}\]

  =1 ,               0 ≤  x  ≤  1

That is, X is uniformly distributed over [O, 1 ].
The marginal pdf of Y is given by

\[h\left( y \right)=\int_{0}^{1}{2\left( x+y-2xy \right)}dx=2\left( xy+\frac{{{x}^{2}}}{2}-y{{x}^{2}} \right)|_{0}^{1}\]

  =1 ,               0 ≤  y  ≤  1

Hence Y is also uniformly distributed over [0, 1 ].

Definition. 
We say that the two-dimensional continuous random variable is uniformly distributed over a region R in the Euclidean plane if
f(x, y) = const                   for (x, y) ϵR,
           = 0,                          elsewhere.

Because of the requirement the above implies  \[\int_{-\infty }^{+\infty }{\int_{-\infty }^{+\infty }{f\left( x,y \right)dydx=1}}\] that the constant equals I/area (R). We are assuming that R is a region
with finite, nonzero area.
Note: This definition represents the two-dimensional analog to the one-dimensional
uniformly distributed random variable.
EXAMPLE 3
. Suppose that the two-dimensional random variable (X, Y) is uniformly distributed over the shaded region R
 f(x, y) = 1/area(R)'                    (x, y)  ϵ R.
We find that
\[area\left( R\right)=\int_{0}^{1}{\left( x-{{x}^{2}} \right)}dx=\frac{1}{6}\]

Therefore the pdf is given by
f(x,y) = 6,                        (x,y) ∉R
          = 0,                        (x, y) ∉. R.

In the following equations we find the marginal pdf's of X and Y.
\[g\left( x \right)=\int_{-\infty }^{+\infty }{f\left( x,y \right)dy=\int_{{{x}^{2}}}^{x}{6dy=6\left( x-{{x}^{2}} \right)}},\]                                              
0 ≤  x  ≤  1
\[h\left( y \right)=\int_{-\infty }^{+\infty }{f\left( x,y \right)dx=\int_{y}^{\sqrt{y}}{6dy=6\left( \sqrt{y}-y \right)}},\]
                                       0 ≤  y  ≤  1

The concept of conditional probability may be introduced in a very natural way.

EXAMPLE 4
Suppose that we want to evaluate the conditional probability P(X = 2 | Y = 2). According to the definition
of conditional probability we have

P(X = 2 | y = 2) ={P(X= 2 , y = 2)} / P(Y = 2)=0.05/ 0.25= 0 20

We can carry out such a computation quite generally for the discrete case. We have
p(x_{i |} y_j) = P(X = x_i |Y = y_j)
             =p(x_{i ,}y_j) /q(y_i)                        if  q(y_j ) > 0

q(y_j | x_i ) = P( y = y_j | x = x_i)
                 =p(x_i , y_j) / p(xi)                 if p(xi) > 0.

Note: For given j, p(x_{i |} y_j) satisfies all the conditions for a probability distribution.
We have 0 ≤  p(x_{i |} y_j) and also

\[\sum\limits_{i=1}^{\infty }{p\left( {{x}_{i}}|{{y}_{j}} \right)=\sum\limits_{i=1}^{\infty }{\frac{p\left( {{x}_{i}}|{{y}_{j}} \right)}{q\left( {{y}_{j}} \right)}}}=\frac{q\left( {{y}_{j}} \right)}{q\left( {{y}_{j}} \right)}=1\]

In the continuous case the formulation of conditional probability presents some
difficulty since for any given x₀ , y₀ we have P(X = x₀ ) = P( Y =y₀ ) = 0. We
make the following formal definitions.

Definition. 
Let (X, Y) be a continuous two-dimensional random variable with joint pdf f Let g and h be the marginal pdrs of X and Y, respectively. The conditional pdf of X for given Y = y is defined by
g(x | y ) = f(x, y)' / h(y)                      h(y) > 0.
The conditional pdf of Y for given X = x is defined by
h(y | x) = f(x,y)' / g(x)                       g(x) > 0.

Notes:
(a) The above conditional pdf's satisfy all the requirements for a one-dimensional
pdf. Thus, for fixed y, we have g(x | y) ≥ 0 and

\[\int_{-\infty }^{+\infty }{g\left( x|y \right)=\int_{-\infty }^{+\infty }{\frac{f\left( x,y \right)}{h\left( y \right)}}}dx=\frac{1}{h\left( y \right)}\int_{-\infty }^{+\infty }{f\left( x,y \right)}dx=\frac{h\left( y \right)}{h\left( y \right)}=1\]

An analogous computation may be carried out for h(y | x).
(b) An intuitive interpretation of g(x | y) is obtained if we consider slicing the surface represented by the joint pdf /with the plane y = c, say. The intersection of the plane with the surface z = f(x, y) will result in a one-Dimensional pdf, namely the pdf of X for Y = c. This will be precisely g(x | c).
(c) Suppose that· (X, Y) represents the height and weight of a person, respectively. Let /be the joint pdf of (X, Y) and let g be the marginal pdf of X (irrespective of Y).

Hence  ∫⁶_5.8 g(x) dx would represent the probability of the event {5.8 ≤ X ≤  6} irrespective

of the weight Y. And ∫⁶_5.8g(x  | 150) dx would be interpreted as P(5.8 ≤ X 6 | Y = 150). Strictly speaking, this conditional probability is not defined in view of
our previous convention with conditional probability, since P(Y = 150) = 0. However,
we simply use the above integral to define this probability. Certainly on intuitive grounds
this ought to be the meaning of this number.