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Unit 15: Exponential Distribution and Normal Distribution

15.2 Uniform Distribution (Continuous Variable) Notes

A continuous random variable X is said to be uniformly distributed in a close interval (a, b) with
probability density function p(X) if

1
p X for a £ X £ b and
( ) =
b a
-
= 0 otherwise.
The uniform distribution is alternatively known as rectangular distribution. The diagram of the
probability density function is shown in the figure 15.1.

Figure 15.1

X

X

Note that the total area under the curve is unity, i.e. ,

b 1 1 b
ò a b a- dX = b a X = 1
a
-
Further, b 1 z b X dX = 1 X 2 b = a + b
E Xg =
.
b -a a b -a 2 2
a
1 z b b 3 -a 3 1 i
2
2
.
E X d i = X dX = g = db 2 +ab a 2
+
b -a a 3b -a 3
b
1 2 2 (a b+ ) 2 (b a- ) 2
( ) =
+
 Var X (b + ab a ) - =
3 4 12
Example 27: The buses on a certain route run after every 20 minutes. If a person arrives
at the bus stop at random, what is the probability that
(a) he has to wait between 5 to 15 minutes,

(b) he gets a bus within 10 minutes,
(c) he has to wait at least 15 minutes.
Solution.
Let the random variable X denote the waiting time, which follows a uniform distribution with
p.d.f.

1
f X for 0 £ X £ 20
( ) =
20

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