162 Chapter 5: Special Random Variables
The mean of a uniform[α,β]random variable is
E[X]=∫βαx
β−αdx=β^2 −α^2
2(β−α)=(β−α)(β+α)
2(β−α)or
E[X]=α+β
2Or, 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∫βαx^2 dx=β^3 −α^3
3(β−α)=β^2 +αβ+α^2
3and so
Var(X)=β^2 +αβ+α^2
3−(
α+β
2) 2=α^2 +β^2 − 2 αβ
12=(β−α)^2
12EXAMPLE 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).