Bivariate Discrete Probability Distribution

Introduction

So far we have learnt what is meant by a discrete random variable, its probability distribution and expectation. Throughout, we assumed that a single characteristic say X, is of interest. For example, X was the number on the face of a die, number of heads when 3 coins tossed, sum of two faces when two dice thrown etc. Hence, the set-up was univariate; concerning a single r.v. However, there are many situations when we are interested in two characteristics, say X and Y at the same time related to the same item. That is, there are two r.v.s. to be observed simultaneously. In other words, X and Y are two r.v.s. defined on the same sample space Q. Following examples illustrate this point.

1. In a family planning survey, number of children (X), as well as number of girls (Y) in a family are recorded.

2.Sex of the person (X) as well as whether he is smoker or not (Y) is recorded.

3.In forestry, diseased plants from two species are identified.

X = 1 if the plant is diseased and X = 0 otherwise.

Y = 1 if the plant of species I and Y = 0 otherwise.

4. X may be the sum of two numbers and Y may be the maximum

of two numbers when two dice are thrown.

5. X may be the sale in kg of tea of Brand A while Y may be the sale of tea of Brand B.

These situations demand handling two variables at a time. Hence, we combine them into an ordered pair say (X, Y) and call it as a bivariate random variable or two dimensional r.v. Note that both X and Y are defined on the same sample space 2. Whenever both X and Y are discrete (finite in this course), we say that (X, Y) is a two-dimensional discrete r.v.

Definition 1: Bivariate (Two Dimensional) discrete r.v.: Let be the sample corresponding to a random experiment. Let X and Y be two real valued functions (r.v.s.) defined on 2. The ordered pair (X, Y) defined by (X, Y) (w) = (X (w), Y (w)) is called a bivariate or two dimensional random variable on Q. If the number of possible values of (X, Y) is finite, then (X, Y) is called as a bivariate discrete r.v.

Note that (X, Y) is discrete if and only if X and Y are both discrete. Hence, if X takes values X1, X2, ..., Xm and Y takes values y1, Y2, ... Yn: then the range space of (X, Y) is

R(x, y) =((xi, Yi); i = 1, 2, ..., m; j = 1, ..., n}

R(x, y) is nothing but the cartesian product of R, and Ry, the ranges of X and Y respectively. R(x, y) contains m × n points.

Bivariate (Joint) Probability Distribution

Recall that the p.m.f. P (x) of a single r.v. X, was derived by adding the probabilities of the elements in which gives rise to the event [X = x]. On similar lines one can construct the joint probability mass function of the bivariate r.v. (X, Y).

Definition 2: Joint Probability Mass Function of (X, Y) : Let (X, Y) be a discrete bivariate r.v. defined on . Let the range space of (X, Y) be

R(x, y) =

For each (xi, yj) =

{(x, y); i = 1,..., m; j = 1, 2, ..., n} R(x, y) we define a function,

P (xi, yj)

Pij as follows:

Pij = P(xi, yj) = P(X = xi, Y = yj];

if, (i) Pij≥ 0 for all i, j

i= 1, 2, ..., m; j = 1, 2, ..., n

(ii) Σ

Σ Ρ = 1.

i=1 j=1

then the above function P is called as the joint probability mass function of (X, Y).

The set {(xi, yj, Pij); i = 1, 2, ..., m; j = 1, 2, ..., n} is called the joint probability distribution of (X, Y); which can be represented in the following tabular form.

P (X = xi, Y = y;] is obtained by adding the probabilities of those elements in which give rise to the event [X = x;] [Y=yj].

Y У1 Y2

Table 2.1 Yj

...

Уп

Total

P11

P12

...

P21

P22

...

Pij

P2i

www

Pin P2n

P2.

The row totals P1., P2, ..., Pm represent marginal probabilities of X while the column totals P1, P2, P.n represent marginal probabilities of Y.

Marginal Probability Distributions

Let {(xi, yj, Pij); i = 1, ..., m; j = 1, ... n} represent a joint probability distribution of a two-dimensional discrete r.v. (X, Y). We know that by pij we mean P (X = X;, Y = y;]. Using these, we can obtain the (univariate) probability distribution of X as follows.

P1 = P(X = x;)

= P(X = x;)]

= P(x=x)

(X n

(x=3}]}]

j=1

{(X = x) ~ (Y = y;}]}]}

= P(X = Xi, Y=

i=1

Σ P(X = X, Y = y;)

using the distributive property

Since, the events (X = xi, Y = yj) for fixed x; and different yj are disjoint.

= Σ Pij j=1

i = 1, 2, ..., m

The above obtained probability distribution is called as the marginal probability distribution of X. It can be presented in the following

tabular form.

X P (X = X;) P1.

...

P2.

Xi Pi

...

Total 1

On similar lines we can derive the marginal probability distribution of Y.

P. j = P(Y = y;)

U (X = Xi,

= 1 1

Pij

i=1

Yı Y2

...

P(Y=yj) P P.2

Obviously,

j = 1, 2, ..., n

Уп

Total

...

P.j

P. n

Σ Pi-

P.;=1

i=1

j=1

Thus, the row totals in Table 2.1 give the probabilities P (X = x;) for various values of x;; while the column totals give the probabilities P(Y= y;) for various values of yj.

Solved Examples

Example: Two fair dice are thrown. Let X denote the absolute difference between the two scores and Y denote the maximum of two

scores.

(i) Derive the joint probability distribution of (X, Y).

(ii) Also obtain the marginal probability distributions of X and Y. Solution We know that contains 36 elements each with probability 36. The possible values of X are 0, 1, 2, 3, 4, 5 and those

of Y are 1, 2, 3, 4, 5, 6. Now, consider P (X = 0, Y = 1). It means that the difference between two numbers is 0 and the maximum between two numbers is 1. This can happen only when (1, 1) occurs. That is, when both the faces show 1.

P(X=0, Y=1) = 36

Similarly, P (X = 0, Y2) = P(2, 2) = 36

P(X=0, Y=6) = P(6,6) = 36

Further, P (X = 1, Y = 1) = Probability that the difference between two numbers is 1, while maximum is 1. This is an impossible event, as no face of the die is marked as 0.

P(X = 1, Y = 1) = 0

Independence of Two Discrete Random Variables

In earlier chapter, we defined the independence of two events. Events A and B are independent if and only if P (AB) = P(A) · P(B). Extending this definition to two discrete r.v.s. X and Y, we say that X and Y are independent if and only if the events [X = x] and [Y = y] are independent for all values of X and Y. The definition in terms of p.m.f.'s is as follows.

Definition 3: Let (X, Y) be a discrete bivariate r.v. defined on a sample space 2. Let pij = P (X = xi, Y = yj); i = 1, ..., m; j = 1, ..., n denote the joint p.m.f. of (X, Y). Let P., i = 1, 2, ..., m and P. j, j = 1, 2, ..., n be the marginal p.m.f. s of X and Y respectively. Then X and Y are called independent r.v.s. if and only if,

In other words,

Pij P. XP.j; i = 1,..., m; j = 1, ..., n.

P[X= xi, Y=yj] = P(X = x;) P(Y= y;) for all i and j.

Conditional Probability Distributions

The conditional probability of event A given event B is defined as P (ANB) P(B)

P(A|B) =

P(B) > 0

Similarly, the conditional probability of (X = x;) given that (Y = y;)

P(X = xi, Y=yj) Pij P(X = x; Y=yj) = P(Y=yj)

When Y is fixed at yj, and we obtain conditional probabilities of each of xi, i = 1, ..., m given the event (Y= y;), what we get the conditional probability distribution of X given Y = yj. It is represented as follows:

X X2

...

P(X = x; Y=yj) Pij

P2j

...

P.j

P. ¡

Xi Pij P.j

...

Pmj P.j

Total 1

When we fix X at a particular value xi, say and obtain the conditional probabilities of Y = y; for all j, given X = x;, then we get

the conditional probability distribution of X given Y = yj. It is represented as follows:

P(X = x; Y=yj)

Pij P2j

Pij

Pmj

Total 1

P.j

P. ¡

When we fix X at a particular value xj, say and obtain the conditional probabilities of Y = y; for all j, given X = x;, then we get the conditional probability distribution of Y given X = Xj.

P(X = xi, Y = yi)

Pij

P(Y=yj|X=x;) =

P (X = X;)

Pi.

It is tabulated as follows.

Yı P(Y=yi|X = x1) Pil Pi2 Pi Pi.

Y2 ...

...

Pij Pi.

Pin Pi.

Total 1

Remark 1: Notice that when we obtain conditional distribution of X given Y = yj, the underlying variable is X and not Y. In fact Y is hold constant at y;. Similarly, in case of conditional distribution of Y given X = xi, the variable is Y and not X. In fact X is hold constant at xj. Accordingly, conditional probability distributions are probability distributions.

univariate

Σ Pij

Pij

Remark 2:

Σ P(X=xY=yj) =

i = 1

i=1

P.j

= 1

and

Σ P(Y=y; X = x;) = Σ

n Pij Pi. Pi. Pi.

= 1

j=1

These conditions have to be fulfilled as these are p.m.f.s.

Remark 3 When X and Y are independent, the conditional probability distributions become nothing but the respective marginal distributions. For,

P(X=x1Y=yj) =

P(X = xi, Y = y;) P(Y=yj)

P(X = x) P(Y= y;) P(Y=yj)

Pi. P.j

= Pi.

for i=1,..., m

P. j

Pij

P. P. j Pi.

= P. j.

for j = 1, 2, ..., n.

Also, P(Y=yj|X= x;) = P1.

This is in consistency with P (A|B) = P(A) when A and B are independent.

Remark 4: For obtaining conditional probability distribution of X given Y = yj; mark the column of Y = y; and to each entry in this column divide by its column total. Similarly, to get the conditional probability distribution of Y given X = x;, mark the ith row and to each value in this row, divide by the row total.

Joint Distribution Function of Two Dimensional Discrete r.v.

Definition 4: Let (X, Y) be a bivariate discrete r.v. defined on Q. The joint (cumulative) distribution function F (x, y) is defined as,

F(x, y) = P(X ≤x, Y≤y],

x, ye R

We state below some of the important properties of the distribution function.

0 ≤ F(x, y) ≤1

The function is non-decreasing in each of the variables. i.e.

(i) F(x, y) ≤ F(x, y')

if y<y'

(ii) F(x, y) F(x, y)

if x < x'

lim

x→→ ∞ F(x, y) = 0.

lim

x→ F(x, y) = 1

y → ∞

Let a, b, c, d be any real numbers with a < b, c <d. Then,

P[a<Xb, c< Y≤d] = F(b, d) + F (a, c) - F (b, c) - F (a, d)

Probability Distribution of (X + Y) when X & Y are Independent

Let (X, Y) be a bivariate r.v. with range space as {(xi, yj), i = 1, 2, ..., m; j = 1, 2, ..., n}. Define Z = X + Y. Let P, (x) and P2 (y) denote the marginal p.m.f.s of X and Y respectively. Consider

Σ PIX=X, Y = z-xi]

P[Z=z] =

i=1

= Σ i=1

P[X=x]P[Y=z-x;]

.. X and Y are independent

Σ P1 (xi) P2 (z-xi)

i=1

DESCRIPTIVE STATISTICS

Attributes,Variables and types of data

Presentation of Data

Measures of Central Tendency