Two-Dimensional Random Variables
In our study of random variables we have, so far, considered only the one dimensionalcase. That is, the outcome of the experiment could be recorded as a
single number x.
In many situations, however, we are interested in observing two or more numerical
characteristics simultaneously. For example, the hardness Hand the tensile
strength T of a manufactured piece of steel may be of interest, and we would consider
(h, t) as a single experimental outcome. We might study the height Hand
the weight W of some chosen person, giving rise Jo the outcome (h, w). Finally
we might observe the total rainfall R and the average temperature T at a certain
locality during a specified month; giving rise to the outcome (r, t).
We shall make the following formal definition.
Definition.
Let e be an experiment and S a sample space associated with e.
Let X = X(s) and Y = Y(s) be two functions each assigning a real number
to each outcomes s ϵ s We call (X, Y ) a two-dimensional random
variable (sometimes called a random vector).
If X1 = X1(s), X2 = X2(s), ... , Xn = Xn(s) are n functions each assigning
a real number to every outcomes s ϵ s , we call (X1. ... , Xn) an n-dimensional
random variable (or an n-dimensional random vector.)
Note: As in the one-dimensional case, our concern will be not with the functional
nature of X(s) and Y(s), but rather with the values which X and Y assume. We shall
again speak of the range space of (X, Y), say RxxY, as the set of all possible values of
(X, Y). In the two-dimensional case, for instance, the range space of (X, Y) will be a
subset of the Euclidean plane. Each outcome X(s), Y(s) may be represented as a point
(x, y) in the plane. We will again suppress the functional nature of X and Y by writing,
for example, P[X ≤ a, Y ≤ b] instead of P[X(s) ≤ a, Y(s) ≤ b].
As in the one-dimensional case, we shall distinguish between two basic types of random
variables: the discrete and the continuous random variables.
Definition.
(X, Y) is a two-dimensional discrete random variable if the possible
values of (X, Y) are finite or countably infinite. That is, the possible values
of (X, Y) may be represented as (xi, Yi), i = l, 2, ... , n, ... ; j = 1, 2, ... ,m, . . .
(X, Y) is a two-dimensional continuous random variable if (X, Y) can assume
all values in some non countable set of the Euclidean plane. [For example, if
(X, Y) assumes all values in the rectangle {(x, y) | a ≤ x ≤ b, c ≤ y ≤ d}
or all values in the circle {(x,y) | x2 + y2 ≤ l} , we would say that (X, Y)
is a continuous two-dimensional random variable.]
Notes:
(a) Speaking informally, (X, Y) is a two-dimensional random variable if it
represents the outcome of ·a random experiment in which we have measured the two
numerical characteristics X and Y.
(b) It may happen that one of the components of (X, Y), say X, is discrete, while
the other is continuous. However, in most applications we deal only with the cases
discussed above, in which either both components are discrete or both are continuous.
(c) In many situations the two random variables X and Y, when considered jointly,
are in a very natural way the outcome of a single experiment, as illustrated in the above
examples. For instance, X and Y may represent the height and weight of the same
individual, etc. However this sort of connection need not exist. For example, X might
be the current flowing in a circuit at a specified moment, while Y might be the temperature
in the room at that moment, and we could then consider the two-dimensional
random variable (X, Y). In most applications there is a very definite reason for considering
X and Y jointly.
We proceed in analogy with the one-dimensional case in describing the probability
distribution of (X, Y).
Definition. (a) Let (X, Y) be a two-dimensional discrete random variable. With
each possible outcome (xi, Yi) we associate a number p(xi, Yi) representing
P(X = Xi, Y = Yi) and satisfying the following conditions:
(1) p(xi, Yi)
(2)\[\sum\limits_{j=1}^{\infty }{\sum\limits_{i=1}^{\infty }{p\left( {{x}_{ij}}{{y}_{j}} \right)=1}}\]
The function p defined for all (xi, yj ) in the range space of (X, Y) is called the
probability function of ( X, Y). The set of triples (xi, yj p(xi, yj)), i, j = I, 2,
... , is sometimes called the probability distribution of (X, Y).
(b) Let (X, Y) be a continuous random variable assuming all values in some
region R of the Euclidean plane. The joint probability density function f is a
function satisfying the following conditions:
(3) f(x, y) ≥ 0 for all (x, y) ϵ R,
(4) ∬Rf(x, y) dx dy = 1.
Notes: (a) The analogy to a mass distribution is again clear. We have a unit mass
distributed over a region in the plane. In the discrete case, all the mass is concentrated
at a finite or countably infinite number of places with mass p(xi, yj) located at (xi, yj).
In the continuous case, mass is found at all points of some noncountable set in the plane.
(b) Condition 4 states that the total volume under the surface given by the equation
z = f(x, y) equals 1.
(c) As in .the one-dimensional case, f(x, y) does not represent the probability of anything.
However, for positive ∆x and ∆Y sufficiently small, f(x, y) ∆x ∆Y is approximately
equal to P(x ≤ X ≤ x + ∆x, y≤ Y ≤ y + ∆y).
(d) As in the one-dimensional case we shall adopt the convention that f(x, y) = 0
if (x, y) f/:. R. Hence we may consider f defined for all (x, y) in the plane and the requirement
4 above becomes∬ f(x, y) dx dy = 1.
(e) We shall again suppress the functional nature of the two-dimensional random
variable (X, Y). We should always be writing statements of the form P[X(s) = xi, Y(s) = yj], etc. However, if our shortcut notation is understood, no difficulty should arise.
(f) Again, as in the one-dimensional case, the probability distribution of (X, Y) is
actually induced by the probability of events associated with the original sample space S.
However, we shall be concerned mainly with the values of (X, Y) and hence deal directly
with the range space of (X, Y). Nevertheless, the reader should not lose sight of the fact
that if P(A) is specified for all events A ⊂ S, then the probability associated with events
in the range space of (X, Y) is determined. That is, if B is in the range space of (X, Y),
we have
P(B) = P[(X(s), Y(s) ϵ B) = P[s | (X(s), Y(s)) ϵ B).
This latter probability refers to an event in S and hence dete rmines the probability of B.
In terms of our previous terminology, B and {s |(X(s), Y(s)) ϵ B} are equivalent events
If B is in the range space of (X, Y) we have
P(B) =∑B∑ p(xi, yj), (3)
if (X, Y) is discrete, where the sum is taken over all indices (i,j) for which (x;, y1) ϵ B. And
P(B)=∬B f(x, y)dxdy (4)
if (X, Y) is continuous.
EXAMPLE 1. Two production lines manufacture a certain type of item. Suppose
that the capacity (on any given day) is 5 items for line I and 3 items for line
II. Assume that the number of items actually produced by either production line
is a random variable. Let ( X, Y) represent the two-dimensional random variable
yielding the number of items produced by line I and line II, respectively. Table 6.1
gives the joint probability distribution of (X, Y). Each entry represents
p(xi, yj) = P(X = xi, Y = yj).
Thus p(2, 3) = P(X = 2, Y = 3) = 0.04, etc. Hence if B is defined as
B = {More items are produced by line I than by line II}
we find that
P(B) = 0.(}l + 0.03 + 0.05 + 0.07 + 0.09 + 0.04 + 0.05 + 0.06
+ 0.08 + 0.05 + 0.05 + 0.06 ,+ 0.06 + 0.05
= 0.75.
EXAMPLE 2.
Suppose that a manufacturer of light bulbs is concerned about
the number of bulbs ordered from him during the months of January and February.
Let X and Y denote the number of bulbs ordered during these two months, respectively.
We shall assume that (X, Y) is a two-dimensional continuous random
| Y\X | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| 0 | 0 | 0.01 | 0.03 | 0.05 | 0.07 | 0.09 |
| 1 | 0.01 | 0.02 | 0.04 | 0.05 | 0.06 | 0.08 |
| 2 | 0.01 | 0.03 | 0.05 | 0.05 | 0.05 | 0.06 |
| 3 | 0.01 | 0.02 | 0.04 | 0.06 | 0.06 | 0.05 |
variable with the following joint pdf
f(x,y) = c if 5000 ≤ x ≤ 10,000 and 4000 ≤ y ≤ 9000,
= 0, elsewhere.
To determine c we use the fact that
\[\int_{-\infty
}^{+\infty }{\int_{-\infty }^{+\infty }{f\left( x,y \right)}}dxdy=1\]
Therefore
\[_{-\infty
}^{+\infty }\int_{-\infty }^{+\infty }{f\left( x,y
\right)}dxdy=\int_{4000}^{2000}{\int_{5000}^{10000}{f\left( x,y
\right)}}dxdy=c{{\left[ 5000 \right]}^{2}}\]
\[Thus,c={{\left( 5000 \right)}^{-2}},hence\_if,B=\left( X\ge Y \right),we\_have\]
\[Thus,c={{\left( 5000 \right)}^{-2}},hence\_if,B=\left( X\ge Y \right),we\_have\]
\[P\left( B
\right)=1-\frac{1}{{{\left( 5000
\right)}^{2}}}\int_{5000}^{9000}{\int_{5000}^{y}{dxdy}}=1-\frac{1}{{{\left(
5000 \right)}^{2}}}\int_{5000}^{9000}{\left[ y-5000 \right]dy=\frac{17}{25}}]\]
.
Note: In the above example, X and Y should obviously be integer-valued since we
cannot order a fractional number of bulbs! However, we are again dealing with the
idealized situation in which we allow X to assume all values between 5000 and 10,000
(inclusive).
EXAMPLE 2
Suppose that the two-dimensional continuous random variable
(X, Y) has joint pdf given by
f(x,y) = x2 + xy/3 0 ≤ x ≤ 1 , 0 ≤ y ≤ 2
= 0, elsewhere
To check that \[\int_{-\infty }^{+\infty }{\int_{-\infty }^{+\infty }{f\left( x,y \right)}dxdy=1}\] \[\int_{-\infty }^{+\infty }{\int_{-\infty }^{+\infty }{f\left( x,y \right)}}dxdy=\int_{1}^{2}{\int_{0}^{1}{(\frac{{{x}^{3}}}{3}+\frac{{{x}^{2}}y}{6})}}dxdy\]
To check that \[\int_{-\infty }^{+\infty }{\int_{-\infty }^{+\infty }{f\left( x,y \right)}dxdy=1}\] \[\int_{-\infty }^{+\infty }{\int_{-\infty }^{+\infty }{f\left( x,y \right)}}dxdy=\int_{1}^{2}{\int_{0}^{1}{(\frac{{{x}^{3}}}{3}+\frac{{{x}^{2}}y}{6})}}dxdy\]
\[=\int_{0}^{2}{\int_{0}^{1}{({{x}^{2}}+\frac{xy}{3}}}dx)dy\] \[=\int_{0}^{2}{(\frac{1}{3}+\frac{y}{6}})dy=\frac{1}{3}y+\frac{{{y}^{2}}}{12}|_{0}^{2}\] \[=\frac{2}{3}+\frac{4}{12}=1,\] \\[[Let\_\mathbf{B}=\left\{ X+Y\ge 1 \right\}\_We\_shall\_compute\_P\left( B \right)\_by\_evaluating,1-P\left( {\bar{B}} \right)\_where\_\bar{B}=\{X+Y<1 1-x="" b="" dx="" dydx="" frac="" int="" left="" limits_="" right="" span="" style="font-family: "calibri" , sans-serif;" x="" xy="">In studying one-dimensional random variables,
played an important role. In the two-dimensional case we can again define a cumulative function as follows.
Definition.
Let (X , Y) be a two-dimensional random variable. The cumulative (cdf) F of the two-dimensional random variable (X, Y) is defined by
F(x , y) = P(X ≤ x, Y ≤ y).
Note:
F is a function of two variables and has a number of properties analogous to
those discussed for the one-dimensional cdf. We shall mention only
the following important property.
If F is the cdf of a two-dimensional random variable with joint pdf f, then
d2F(x, y)/dxdy = f(x, y)
wherever F is differentiable. which we proved that (d/ dx)F(x) = f(x), where f is the pdf of the one-dimensional random variable X.
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