Continuous Bivariate Distributions

Objectives

• To find the marginal probability distribution and conditional probability distributions.
• Identify whether the variables are independent.
• To find the joint moments and M.G.F.
• To find probabilities of joint events of X and Y.
• To find correlation and regression lines.
• To find the probability distributions of functions of (X, Y).

Introduction

Suppose there is an uncountably infinite sample space. We can associate two real-valued functions with it. For example, in a manufacturing process, inspection of springs is conducted. We record the inner diameter as well as the tensile strength of each spring. Thus, with respect to every sample point, we record values of two jointly variable quantities. The joint study of such variables needs a bivariate probability distribution to be introduced. Moreover, we require finding the distribution of the sum of two or more random variables. In this situation, we need to use bivariate or multivariate probability distributions. In this chapter, we particularly study bivariate continuous types of probability distributions.

Continuous Bivariate Random Variable

Definition

Let be the sample space corresponding to a random experiment. Let, X(w) and Y(w) be two real-valued continuous functions with domain . Then the ordered pair (X(@), Y()) is called a bivariate or two-dimensional continuous random variable, w being a sample point in 2.

Note:

1. For simplicity, we write X and Y in place of X(@) and Y(@) respectively.
2. If X has range set A ⊆ R and Y has range set B ⊆ R then, (X, Y):

Illustrations

1. X(0) = Weight of wth student.
2. Y(0) = Height of wth student.
3. X(w) = pH value (acidity) of water at a pond at time t, Y(0) = Growth of fungal plant at the same time.
4. X(w) = Distance of a space denoted by w from the equator, Y(w) = Temperature at the same place.

Joint Probability Density Function

In the case of a continuous random variable, probability mass is not attached to any individual value. If (X, Y) is a bivariate random variable, then probability is attached to a two-dimensional region, such as {(x, y) | a < x < b, c < y < d}. The probability varies from region to region. In order to obtain the probability mass associated with any two-dimensional region, we need to take into account the concept of the probability density function. A bivariate function f(x, y) can be treated as a probability density function if it is a non-negative and continuous function of (x, y).

The probability that a r.v. (X, Y) takes values in a small region SxSy < x < x + Δx, < y < y + Δy will be the product of the area of the region and the value of f(x, y) in the region:

P(SxSy) = f(x, y)ΔxΔy.

Thus, the above probability is the volume of a box with dimensions f(x, y), Δx, and Δy.

Definition

A function f(x, y) is called a joint probability density function of a bivariate continuous r.v. (x, y) if:

1. f(x, y) = 0 for -∞ < x, y < ∞
2. ∫∫ f(x, y) dx dy = 1.

Note:

1. P(a < x < b, c < y < d) = ∫∫ f(x, y) dx dy.
2. The total volume under surface f(x, y) which is bounded by the xy-plane is 1.
3. If a, b, c, d are constants, then ∫∫ f(x, y) dx dy = ∫∫ f(x, y) dy dx.

Double Integrals

While dealing with continuous random variables, we need to evaluate double integrals many times. We summarize the required results in the subsequent discussion.

1. To evaluate ∫∫ f(x, y) dx dy, first we evaluate the inner integral with respect to x, and then the outer integral with respect to y. The order can be interchanged.
2. If the limits of integration with respect to y are functions of x and those of x are constant, then the inner integral is with respect to y and vice versa.
3. If A is a region of integration given by A = {(x, y) | a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)}, the order of the integral can be interchanged.

Distribution Function

Definition

If (X, Y) is a bivariate r.v. with joint p.d.f. f(x, y), then the joint distribution function is denoted by F(x, y) and defined as:

F(x, y) = P(X ≤ x; Y ≤ y) = ∫∫ f(u, v) dv du.

Properties of Distribution Function F(x, y)

1. Non-negative: F(x, y) ≥ 0 for all (x, y).
2. Non-decreasing in each argument: If x1 < x2, then F(x1, y) ≤ F(x2, y) and if y1 < y2, then F(x, y1) ≤ F(x, y2).
3. 1. F(-∞, -∞) = 0
   2. F(∞, ∞) = 1
   3. F(x, ∞) = F(x)
   4. F(-∞, y) = 0
4. Right-Continuity: F(x, y) is right continuous in each of its arguments.
5. If a < b and c < d, then ΔF(x, y) = F(b, d) - F(a, d) - F(b, c) + F(a, c) ≥ 0.

Marginal Probability Distributions

If (X, Y) is a continuous bivariate r.v. with joint p.d.f. f(x, y), then the probability distribution of X alone is called the marginal probability distribution of X and the probability distribution of Y alone is called the marginal probability distribution of Y.

The marginal p.d.f. of x is given by:

f1(x) = ∫ f(x, y) dy.

The marginal p.d.f. of y is given by:

f2(y) = ∫ f(x, y) dx.

Thus, to get the marginal p.d.f. of a variable we integrate the joint p.d.f. over the range of the other variable.

Conditional Probability Distributions

If a continuous r.v. (X, Y) has joint p.d.f. f(x, y), then the conditional p.d.f. of X given Y = y is:

g1(x | Y = y) = f(x, y) / f2(y).

The conditional p.d.f. of Y given X = x is:

g2(y | X = x) = f(x, y) / f1(x).

Note:

1. The conditional distribution of one variable given the other is a one-dimensional p.d.f. and the same results hold as in the case of one-dimensional p.d.f.
2. The conditional p.d.f. depends on the joint p.d.f. of (X, Y) and the marginal p.d.f. of the given variable.

Independence of Random Variables

Two continuous r.v.s X and Y are said to be independent if for any two real-valued functions φ and ψ, the joint expectation equals the product of the expectations:

E[φ(X)ψ(Y)] = E[φ(X)]E[ψ(Y)].

If (X, Y) has joint p.d.f. f(x, y), then X and Y are independent if and only if:

f(x, y) = f1(x)f2(y) for all (x, y).

where f1(x) and f2(y) are marginal p.d.f.s of X and Y respectively.

Joint Moments and Moment Generating Function

The expectation of the r.v. h(X, Y) is:

E[h(X, Y)] = ∫∫ h(x, y) f(x, y) dx dy.

The moment generating function M(t1, t2) of (X, Y) is:

M(t1, t2) = E[e^(t1X + t2Y)] = ∫∫ e^(t1x + t2y) f(x, y) dx dy.

Joint Moments

The joint moment of (X, Y) about the origin is:

E(X^r Y^s) = ∫∫ x^r y^s f(x, y) dx dy.

The joint central moment is:

E[(X - μx)^r (Y - μy)^s] = ∫∫ (x - μx)^r (y - μy)^s f(x, y) dx dy.

Note:

1. The expectation and moments can be evaluated in a similar way as in the case of one-dimensional random variables.
2. The expectation of the sum of two variables is the sum of their expectations.

Covariance and Correlation

Definition

If (X, Y) is a bivariate random variable, the covariance of X and Y is:

Cov(X, Y) = E[(X - μx)(Y - μy)] = E[XY] - μxμy.

Note:

1. If X and Y are independent, then Cov(X, Y) = 0, but the converse is not necessarily true.
2. Cov(X, X) = Var(X).
3. Cov(X, Y) may be positive or negative.

Correlation Coefficient

The correlation coefficient ρ between X and Y is:

ρ = Cov(X, Y) / (σxσy).

Note:

1. -1 ≤ ρ ≤ 1.
2. If ρ = 0, X and Y are uncorrelated.

Regression Lines

The line of regression of X on Y is:

X - μx = ρ(σx/σy)(Y - μy).

The line of regression of Y on X is:

Y - μy = ρ(σy/σx)(X - μx).

Note:

1. The regression lines give the best linear relationship between the variables.
2. The slope of the regression line depends on the correlation coefficient.

Distribution of Functions of (X, Y)

If X and Y are two continuous random variables with joint p.d.f. f(x, y), we often need to find the distribution of a function of X and Y, say Z = h(X, Y). This can be done using the method of transformations or the method of distribution functions.

Method of Transformations

If Z = h(X, Y), we find the joint p.d.f. of (X, Y) and transform it to get the p.d.f. of Z.

Method of Distribution Functions

If Z = h(X, Y), we find the distribution function of Z, and then differentiate it to get the p.d.f. of Z.