30.4 RANDOM VARIABLES AND DISTRIBUTIONS
l (^1) abl 2
x
f(x)
Figure 30.7 The probability density function for a continuous random vari-
ableXthat can take values only between the limitsl 1 andl 2. The shaded area
under the curve gives Pr(a<X≤b), whereas the total area under the curve,
between the limitsl 1 andl 2 , is equal to unity.
i.e.f(x)dxis the probability thatXlies in the intervalx<X≤x+dx. Clearly
f(x) must be a real function that is everywhere≥0. IfXcan take only values
between the limitsl 1 andl 2 then, in order for the sum of the probabilities of all
possible outcomes to be equal to unity, we require
∫l 2
l 1
f(x)dx=1.
OftenXcan take any value between−∞and∞and so
∫∞
−∞
f(x)dx=1.
The probability thatXlies in the intervala<X≤bis then given by
Pr(a<X≤b)=
∫b
a
f(x)dx, (30.42)
i.e. Pr(a<X≤b) is equal to the area under the curve off(x) between these
limits (see figure 30.7).
We may also define the cumulative probability functionF(x) for a continuous
random variable by
F(x)=Pr(X≤x)=
∫x
l 1
f(u)du, (30.43)
whereuis a (dummy) integration variable. We can then write
Pr(a<X≤b)=F(b)−F(a).
From (30.43) it is clear thatf(x)=dF(x)/dx.