10.3 The Uniform Distribution

Let X be an rv with pdf f _x (x). The rv X is said to be uniformly distributed in the interval a ? x ? b if

The pdf of the uniform distribution is shown in Figure 10.3.

Figure 10.3: The pdf of the uniform distribution.

From Figure 10.3 we observe that

and

The cdf of the uniform distribution if found from

and it is shown in Figure 10.4.

Figure 10.4: The cdf of the uniform distribution

We observe that in Figure (10.4),

and thus the equation of the straight line of the cdf, in terms of the slope and the vertical axis intercept c (not shown), is

To find the intercept c, we evaluate P _x(x) at x = a. Then,

Therefore,

and by substitution (10.21) into (10.20), we obtain

Therefore, the cdf of the uniform distribution for the entire interval 0 ? x ? ? is

Example 10.4

A Silicon Valley commuter has determined that the commuting time from home to work varies between 55 and 75 minutes. Compute the probability that in any day, the commuting time will be between 60 and 70 minutes.

Solution:

The pdf of the uniform distribution for this example is shown in Figure 10.5 where b = 75 and a = 55.

Figure 10.5: The pdf of Example 10.4

Then, b ? a = 20, and 1/(b ? a) = 1/20 = 0.05. The probability that...