162 Chapter 5: Special Random Variables
The mean of a uniform[α,β]random variable is
E[X]=
∫β
α
x
β−α
dx
=
β^2 −α^2
2(β−α)
=
(β−α)(β+α)
2(β−α)
or
E[X]=
α+β
2
Or, in other words, the expected value of a uniform[α,β]random variable is equal to the
midpoint of the interval[α,β], which is clearly what one would expect. (Why?)
The variance is computed as follows.
E[X^2 ]=
1
β−α
∫β
α
x^2 dx
=
β^3 −α^3
3(β−α)
=
β^2 +αβ+α^2
3
and so
Var(X)=
β^2 +αβ+α^2
3
−
(
α+β
2
) 2
=
α^2 +β^2 − 2 αβ
12
=
(β−α)^2
12
EXAMPLE 5.4c The current in a semiconductor diode is often measured by the Shockley
equation
I=I 0 (eaV−1)
whereVis the voltage across the diode;I 0 is the reverse current;ais a constant; andIis
the resulting diode current. FindE[I]ifa=5,I 0 = 10 −^6 , andVis uniformly distributed
over (1, 3).