Probability Density Functions

Quantities that range continuously over an interval of real numbers are called continuous random variables and every continuous random variable has a probability density function with the property that the probability of lying between the numbers and is given by the integral

In general, for all and since the value of is always some real number, it follows that so

In geometric terms, the proability is the area under the graph of over the interval
If and are both continuous random variables, then the joint probability density function for two random variables and is a function of two variables such that for all and

where denotes the probability that is in the region Note that

Geometrically, may be thought of as the volume under the surface above the region .

Example (Probability Density Functions) Suppose the joint probability density function for the random variable and is is modeled by

Find the probability that

Solution. The probability that is given as

Let and then and and so using integration by parts we find,

Example (Probability Density Functions) Suppose the joint probability density function for the random variable and is is modeled by

Find the probability that

Solution. We have

Example (Probability Density Functions) Suppose measures the time (in minutes) that a person stands in line at a certain bank and the duration (in minutes) of a routine transaction at the teller's window. You arrive at the bank to deposit a check. The joint probability density function for and is modeled by

Find the probability that you will complete your business at the bank within 8 minutes.

Solution. The probability that you will complete your business at the bank within 8 minutes is