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Distribution of Several
Random Variables
Distribution of Several Random Variables
• Distributions of two or more random variables are of interest for
two reasons:
• 1. They occur in experiments in which we observe several
random variables, for example carbon content X and hardness
Y of steel, amount of fertilizer X and yield of corn Y, height X1,
weight X2, and blood presure X3 of persons and so on.
• 2. they will be needed in the mathematical justification of the
methods of statistics.
Distribution of Several Random Variables
• In this section we consider two random variables X and Y or as we
also say a two dimensional random variable (X,Y).
• For (X,Y) the outcome of a trial is a pair of numbers X=x, Y=y
briefly (X,Y) = (x,y) which we can plot as a point in the XY-plae.
• The two dimensional probability distribution of the random
variabe (X,Y) is given by the distribution function
Distribution of Several Random Variables
• This is the probability that in trial, X will assume any value not
greater than x and in the same trial, Y will assume any vale not
greater than y.
• This corresponds to the blue region in he figüre which extends to
-∞ to the left and below.
Distribution of Several Random Variables
• F(x,y) determines the probability distribution uniquely, because in
analogy to formula (2) that is P(𝑎 < X ≦ b ) = F(b) – F(𝑎), we now
have for a rectangle
• As before in the two-dimensional case we shall also have discrete
and continuous random variables and distributions.
Discrete Two-Dimensional Distributions
• In analogy to the case of a single randm variable we call (X.Y) and
its distribution discrete if (X,Y) can asume only finitely many or at
most countably infinitely many pairs of values (x1 ,y1), (x2 ,y2), …
with positive probabilities, whereas the probability for any domain
containing none of those values of (X,Y) is zero.
• Let (xi ,yj) be any of those pairs and let P(X= xi, Y= yj) =pij (where
we admit that pij may be 0 fr certain pairs of subscript i,j).
Discrete Two-Dimensional Distributions
• Then we define the probability function f(x,y) of (X,Y) by
• Here, i=1,2,… and j=1,2,… independently. In analogy we now have
for the distribution function the formula
Discrete Two-Dimensional Distributions
• Instead of the previous fomula (6 in slide 5) we now have the
condition
Example
• If we have simultaneously toss a dime and a nickel and consider
X= Number of heads the dime tuns up,
Y= Number of heads the nickel turns up,
Then X and Y can have the values 0 or 1, and the probability function is
f(0,0)=f(1,0)=f(0,1)=f(1,1) = ¼, f(x,y)=0 otherwise.
Continuous Two-Dimensional Distribution
• In analogy to thecase of a single random varible in slide 5 we call
(X,Y) and its distribution continuous if the corresponding
distribution function F(x,y) can be given by a double integral
• Whose integrand f, called the density of (X,Y) is nonnegative
everywhere and is continuous possibly except on finitely many
curves.
Continuous Two-Dimensional Distribution
• From (6) we obtain the probability that (X,Y) assume any value in
the rectangle in Discrete Two-Dimensional Distribtions given by the
formula
Example
• Let R be the rectangle 𝛼1 < x ≦ 𝛽1 , 𝛼2 < x ≦ 𝛽2 . The density is
• Defines the so-called uniform distribution in the rectangle R; here
k=(𝛽1 -𝛼1 )(𝛽2 -𝛼2 ) is the area of R. Density function and distribution
functions are shown.
Marginal Distributions of a Discrete Distribution
• This is rather a natural idea, without counterpart for a single
random variable.
• It amounts to being interested only in one of the two variables in
(X,Y),say X and aking for its distribution called the marginal
distribution of X in (X,Y).
• So we ask for the probability P(X=x, Y arbitrary).
• Since (X,Y) is discrete, so is X.
Marginal Distributions of a Discrete Distribution
• We get its probability function, call it f1(x), from the probability
function f(x,y) of (X,Y) by summing over y:
• Where we sum all the values of f(x,y) that are not 0 for that x.
Marginal Distributions of a Discrete Distribution
• From (9) we see that the distribution function of the marginal
distribution of X is
• Similarly the probability function
• Determines the marginal function of Y in (X,Y).
Marginal Distributions of a Discrete Distribution
• Here we sum all the values of f(x,y) that are not zero for the
corresponding y.
• The distribution function of this marginal distribution is
Example
• In drawing 3 cards with replacement from a bridge deck let us
consider
(X,Y), X= Number of Queens,
Y= Number of Kings or Aces
• The deck has 52 cards. These include 4 queens, 4 kings and 4 aces.
• Hence in a single trial a queen has probability 4/52=1/13 and a
king or ace is 8/25=2/13.
Example
• This gives the probability function of (X,Y),
• And (x,y) =0 otherwise. Next table shows in center the values of
f(x,y) and on the right and lower margins the values of the
probability functions f1(x) and f2(x) of the marginal distributions of
X and Y respectively.
Example
Marginal Distribution of a Continuous Distribution
• This is conceptually same as for the discrete distributions, with probability
functions and sums raplaced by densities and integrals.
• For a continuous random variable (X,Y), defined by the distibution
function
• With the density f1 of X obtained from f(x,y) by intergration over y.
Marginal Distribution of a Continuous Distribution
• Interchanging the roles of X and Y, e obtained the marginal
distribution of Y in (X,Y) with the distribution function
• And density
Independence of Random Variables
• X and Y in a (discrete or continuous) random variable (X,Y) are
said to be independent if
• Holds for all (x,y).
• Otherwise these random variables are said to be dependent.
Independence of Random Variables
• These definitions are suggested by the corresponding definitions
for events in slide 3.
• Necessary and sufficent for independence is
• For all x and y.
• Here the f’s are the above probability functions is (X,Y) is discrete
or those densities if (X,Y) is continuous.
Example
• In tossing a dime and a nickel, X=Number of heads on the dime, Y=
Number of heads on the nickel may assume that the values 0 or 1
and are independent.
• The random variables in the previous table are dependent.
Extensions of Independence
to n-Dimensional Random Variables
• This will be needed throughoutin the next chapter.
• The distribution of such a random variabe X = (X1 ,…,Xn) is
determined by a distribution function of the form
• The random variables X1 ,…,Xn are said to be independent if
• For all (x1, … , xn).
Extensions of Independence
to n-Dimensional Random Variables
• Here Fj(xj) is the distribution function of the marginal distribution
of Xj in X, that is
• Otherwise these random variables are said to be dependent.
Functions of Random Variables
• When n=2, we write X1 =X, X2 =Y, x1=x, x2=y.
• Taking a nonconstant continuos function g(x,y) defined for all x,y,
we obtain a random variable Z=g(X,Y).
• For example, if we roll two dice and X and Y are the numbers the
dice turn up in a trial, then Z=X+Y is the sum of those two
numbers.
Functions of Random Variables
• In the case of a discrete random vriable (X,Y) we may obtain the
probability function f(z) of Z=g(X,Y) by summing all f(x,y) for
which g(x,y) equals the value of z considered, thus
• Hence the distribution function of Z is
• Where we sum all values of f(x,y) for which g(x,y) ≦ z.
Functions of Random Variables
• In the case of a continuous random variable (X,Y) we similary have
• Where for each z we integrate the density f(x,y) of (X,Y) over the
region g(x,y) ≦ z in the xy-plane, the boundry curve of this region
being g(x,y) ≦ z.
Addition of Means
• The number
is called the mathematical expectation or briefly the expectation of
g(X,Y).
Addition of Means
• Here it is assumed that the double series converges absolutely and
the integral of |g(x,y)|f(x,y) overthe xy-plane exists(is finite).
• Since summation and integration are linear processes, we have
from (23)
• An important special case is
• And by induction we have the following result.
Addition of Means
• The mean (expectation) of a sum of random variables equals the
sum of the means (expectations) that is,
• Furthermore, we readily obtain
Multiplcation of Means
• The mean (expectation) of the product of independent random
variables equals the product of the means (expectations), that is,
Proof
• If X and Y are independent random variables (both discrete or
both continuous), then E(XY) = E(X)E(Y).
• In fact in the discrete case we have
• And in the continuous case the proof of the relation is similar.
• Extension to n indeendent random variables gives (26) and
theorem is proved.
Addition of Variances
• This is the another matter of practical şmortance hat we shall
need.
• As before, let Z=X+Y and denote the mean and variance of Z by 𝜇
and 𝜎 2 .
• From (24) we see that the first term on the right equals
Addition of Means
• For the second term on the right we obtain theorem 1
• By substituting these expressions into the formula for 𝜎 2 we have
Addition of Means
• We see that the expression in the first line on the right is the sum
of the variance of X and Y, which denote by 𝜎12 and 𝜎22 ,
respectively
• An is called the covariance of X and . Consequently our result is
Addition of Means
• If X and Y are independent, then
• Hence 𝜎𝑋𝑌 = 0 and
Addition of Means
• Extension to more than two variables gives the basic
• The variance of the sum of independent random variables equals
the um of the variances of these variables.