3.1. The probabilistic facts of life[[Student version, December 8, 2002]] 67
Figure 3.1: (Metaphor.) Examples of intermediate outcomes not allowed in a discrete probability distribution.
[Cartoon by Larry Gonick, from (Gonick & Smith, 1993).] [Copyrighted figure; permission pending.]
3.1.2 Continuous distributions
More oftenxcan take on any value in a continuous intervala≤x≤b.Inthis case, we partition
the interval intobinsof width dx. Again we imagine making many measurements and drawing a
histogram, finding that dN(x 0 )ofthe measurements yield a value forxsomewhere betweenx 0 and
x 0 +dx.Wethen say that the probability of observingxin this interval isP(x 0 )dx,where
dN(x 0 )/N→P(x 0 )dx for largeN. (3.3)Strictly speaking,P(x)isonly defined for the discrete values ofxdefined by the bins. But if we
make enough measurements, we can take the bin widths dxto be as small as we like and still have
alot of measurements in each bin, dN(x)1. IfP(x)approaches a smooth limiting function as
wedo this, then we sayP(x)isthe probability distribution (or probability density) forx. Once
again,P(x)must always be nonnegative.
Equation 3.3 implies that a continuous probability distribution has dimensions inverse to those
ofx.Adiscrete distribution, in contrast, is dimensionless (see Equation 3.1). The reason for this
difference is that the actual number of times we land in a small bin depends on the bin width dx.
In order to get a quantityP(x)that is independent of bin width, we needed to divide dN(x 0 )/N
by dxin Equation 3.3; this operation introduced dimensions.
What if the interval isn’t small? The probability of findingxis then just the sum of all
the bin probabilities making up that interval, or
∫x 2
x 1 dxP(x). The analog of Equation 3.2 is the
normalization condition for a continuous distribution:
∫b
adxP(x)=1. (3.4)Dull Example:Theuniform distributionis a constant from 0 toa:
P(x)={
(1/a),if 0 ≤x≤a;
0,otherwise.(3.5)
Interesting Example:The famousGaussian distribution,or“bell curve,” or “normal distribution”
is
P(x)=Ae−(x−x^0 )
(^2) / 2 σ 2
, (3.6)