Introduction to Probability and Statistics for Engineers and Scientists

5.4The Uniform Random Variable 161

f(x)

a b

1

b – a

ab x

FIGURE 5.5 Probabilities of a uniform random variable.

is a subinterval of[α,β](see Figure 5.5),

P{a<X<b}=

1

β−α

∫b

a

dx

=

b−a

β−α

EXAMPLE 5.4a IfXis uniformly distributed over the interval [0, 10], compute the
probability that(a) 2 <X<9,(b) 1 <X<4,(c)X<5,(d)X>6.

SOLUTION The respective answers are(a)7/10,(b)3/10,(c)5/10,(d)4/10. ■

EXAMPLE 5.4b Buses arrive at a specified stop at 15-minute intervals starting at 7A.M.That
is, they arrive at 7, 7:15, 7:30, 7:45, and so on. If a passenger arrives at the stop at a time
that is uniformly distributed between 7 and 7:30, find the probability that he waits

(a)less than 5 minutes for a bus;

(b)at least 12 minutes for a bus.

SOLUTION LetXdenote the time in minutes past 7A.M. that the passenger arrives at the
stop. SinceXis a uniform random variable over the interval (0, 30), it follows that the
passenger will have to wait less than 5 minutes if he arrives between 7:10 and 7:15 or
between 7:25 and 7:30. Hence, the desired probability for(a)is

P{ 10 <X< 15 }+P{ 25 <X< 30 }= 305 + 305 =^13

Similarly, he would have to wait at least 12 minutes if he arrives between 7 and 7:03 or
between 7:15 and 7:18, and so the probability for(b)is

P{ 0 <X< 3 }+P{ 15 <X< 18 }= 303 + 303 =^15 ■