1 Signal Bars

EXAMPLE 5-1 Signal Bars

  Calls are made to check the airline schedule at your departure

  In the first four bits transmitted, let

  city. You monitor the number of bars of signal strength on your

  X denote the number of bars of signal strength

  cell phone and the number of times you have to state the name

  on your cell phone

  of your departure city before the voice system recognizes the

  Y denote the number of times you need to state

  name.

  your departure city

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  CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

  x = number of bars of signal strength y = number of times city name is stated 1 2 3

  Figure 5-1 Joint

  probability distribution

  of X and Y in Example

  By specifying the probability of each of the points in Fig. 5-1,

  range of the random variables 1 2 to be the set of points X, Y

  we specify the joint probability distribution of X and Y.

  1 x, y 2 in two-dimensional space for which the probability that

  Similarly to an individual random variable, we define the

  X x and Y y is positive.

  If X and Y are discrete random variables, the joint probability distribution of X and Y is a description of the set of points 1x, y2 in the range of 1X, Y 2 along with the probability of each point. 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. Also, P 1X x and Y y2 is usually written as P 1X x, Y y2.

  Joint Probability

  The joint probability mass function of the discrete random variables X and Y,

  Mass Function

  denoted as f XY 1x, y2, satisfies

  (1) f XY 1x, y2 0

  (2) a a f XY 1x, y2 1

  x

  y

  (3) f XY 1x, y2 P1X x, Y y2 (5-1)

  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 specified.

  The joint probability distribution of two continuous random variables X and Y can be specified 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 defined over

  two-dimensional space. The double integral of f XY 1x, y2 over a region R provides the proba- bility that 1X, Y 2 assumes a value in R. This integral can be interpreted as the volume under the

  surface f XY 1x, y2 over the region R.

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

  that 1X, Y 2 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 probabil- ities for X and Y.

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  5-1 TWO OR MORE RANDOM VARIABLES

  Figure 5-2 Joint probability density function for

  Figure 5-3 Joint probability density function for the lengths

  random variables X and Y. Probability that (X, Y ) is

  of different dimensions of an injection-molded part.

  in the region R is determined by the volume of

  f XY (x, y) over the region R.

  Joint Probability

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

  Density

  denoted as f XY

  1x, y2, satisfies the following properties:

  Function

  (1) f XY 1x, y2 0 for all x, y

  f XY 1x, y2 dx dy 1 冮 冮