5-1.1 JOINT PROBABILITY DISTRIBUTIONS

Joint Probability Mass Function

For simplicity, we begin by considering random experiments in which only two random vari- ables are studied. In later sections, we generalize the presentation to the joint probability distribution of more than two random variables.

Mobile Response Time The response time is the speed of page downloads and it is critical for a mobile Web site. As the response time increases, customers become more frustrated and poten- tially abandon the site for a competitive one. Let X denote the number of bars of service, and let Y denote the response time (to the nearest second) for a particular user and site.

Example 5-1

Section 5-1/Two or More Random Variables 157

If X and Y are discrete random variables, the joint probability distribution of X and Y is a descrip- tion of the set of points ( , ) x y in the range of ( , ) X Y along with the probability of each point. Also, P X ( = x and Y = y ) is usually written as P X ( = x Y , = y ) . The joint probability distribution of two random variables is sometimes referred to as the bivariate probability distribution or bivariate

distribution of the random variables. One way to describe the joint probability distribution of two

discrete random variables is through a joint probability mass function f x y ( , ) = P X ( = x Y , = y ) .

The

joint probability mass function

of the discrete random variables X and Y , denoted as f

xy

( , ), satisfi es x y

(1) f

XY

( ) x, y

^$

⁰ (2) f

XY

x, y

Y

X

∑ ( )

∑

⁵

¹

(3) f

XY

( ) x, y

^{5 5}

P X ( x,Y

⁵

y ) ^(5-1)

Joint Probability Mass Function

Just as the probability mass function of a single random variable X is assumed to be zero at all values outside the range of X, so the joint probability mass function of X and Y is assumed to be zero at values for which a probability is not specifi ed.

Joint Probability Density Function

The joint probability distribution of two continuous random variables X and Y can be speci- fi ed by providing a method for calculating the probability that X and Y assume a value in any region R of two-dimensional space. Analogous to the probability density function of a single continuous random variable, a joint probability density function can be defi ned over two- dimensional space. The double integral of f

XY

( , ) over a region R provides the probability x y that ( X Y , ) assumes a value in R . This integral can be interpreted as the volume under the surface f

XY

( ) x y , over the region R .

A joint probability density function for X and Y is shown in Fig. 5-2. The probability that X Y ,

( ) assumes a value in the region R equals the volume of the shaded region in Fig. 5-2. In this manner, a joint probability density function is used to determine probabilities for X and Y .

Typically, f

XY

( ) x y , is defi ned over all of two-dimensional space by assuming that f

XY

( ) x y ,

⁵

0 for all points for which f

XY

( ) x y , is not specifi ed.

FIGURE 5-1 Joint probability

distribution of X and Y in Example 5-1.

x = Number of Bars of Signal Strength y = Response time

(nearest second)

1 2 3

4 0.1 0.05

0.05 0.2 0.25

2 3

0.1 0.03 0.02 0.15

1

0.02 0.02 0.01

By specifying the probability of each of the points in Fig. 5-1, we specify the joint probability distribution of X and Y . Similarly to an individual random variable, we defi ne the range of the random variables ( , ) X Y to be the set of points ( , ) x y in two-dimensional space for which the probability that X = x and Y = y is positive .

A joint probability density function for the continuous random variables X and Y , denoted as f

XY

( ) x, y , satisfi es the following properties:

(1) f

XY

( ) x, y

^$

^{0 for all} x, y

Joint Probability

Density Function

f_XY(x, y)

x y

R

FIGURE 5-2 Joint probability density function for random variables X and Y. Probability that (X,Y) is in the region R is determined by the volume of f x yXY( , ) over the region R.

FIGURE 5-3 Joint probability density function for the lengths of different dimensions of an injection-molded part.

f_XY(x, y)

y

x

3.0 2.95 7.70 3.05

7.80

7.60

At the start of this chapter, the lengths of different dimensions of an injection-molded part were presented as an example of two random variables. However, because the meas- urements are from the same part, the random variables are typically not independent. If the specifi cations for X and Y are [2.95, 3.05] and [7.60, 7.80] millimeters, respectively, we might be interested in the probability that a part satisfi es both specifi cations; that is, P ( 2 95 .

, ,

X 3 05 7 60 . , .

, ,

Y 7 80 . ) . Suppose that f

XY

( ) x y , is shown in Fig. 5-3. The required probability is the volume of f

XY

( ) x y , within the specifi cations. Often a probability such as this must be determined from a numerical integration.

(2) ∫ ∫ f

XY

( ) x, y dx dy

⁵

2 2

1

∞

∞

∞

∞

(3) For any region R of two-dimensional space, P X,Y

R

f

XY

x, y dx dy

( ) ^∈

R

( )

⁵

^{∫∫ ( ) (5-2)}

Server Access Time Let the random variable X denote the time until a computer server con- nects to your machine (in milliseconds), and let Y denote the time until the server authorizes you as a valid user (in milliseconds). Each of these random variables measures the wait from a common starting time and X < Y . Assume that the joint probability density function for X and Y is

f

XY

( ) x, y

^{5 3}

^{6 10}

²⁶

^exp (

²

^{0 001 . x}

²

^{0 002 . y} )     for     x < y

Reasonable assumptions can be used to develop such a distribution, but for now, our focus is on only the joint prob- ability density function.

The region with nonzero probability is shaded in Fig. 5-4. The property that this joint probability density function inte- grates to 1 can be verifi ed by the integral of f

XY

( , ) over this region as follows: x y

f

XY

x, y dy dx e

x y

dy d

( ) ^⎛ _⎝⎜

x

^⎞ _⎠⎟

∫

∫ ∫

2 2

2 2 2

5 3

∞

∞

∞

∞ ∞

6 10

^{6 0 001. 0 002.}

xx e dy e dx

e

y x

0

6 0 002

0

0 001 0

6 0 0

6 10 6 10

∞ ∞ ∞

∫

^{5 3}

∫ ^⎛ _⎝⎜ ∫

⁻

^⎞ _⎠⎟

⁻

5 3

2

2 2

. .

. 002

0 001 0

0 003

0 002 0 003

0

0

x

x x

e dx e dx

.

^.

.

^.

.

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜

⎞

∫

^{2 5}

∫

²

⎠⎟

⁵

∞ ∞

0

003 1

0 003 1

⎛ ⎝⎜ ⎞

⎠⎟

.

5

Example 5-2

Section 5-1/Two or More Random Variables 159

FIGURE 5-5 Region of integration for the probability that X

<

1000 and Y

<

2000 is darkly shaded.

y

x 0

FIGURE 5-4 The joint probability density function of X and Y is nonzero over the shaded region.

y

x 0

0 2000

1000