3.1. The probabilistic facts of life[[Student version, December 8, 2002]] 69
fartherawayfromx 0 before the factor e−(x−x^0 )
(^2) / 2 σ 2
begins to hurt. Remembering thatP(x)isa
probability distribution, this observation means that for biggerσyou’re likely to find measurements
with bigger deviations from the most-likely valuex 0 .The prefactor of 1/σin front of Equation 3.8
arises because a wider bump (largerσ)needs to be lower to maintain a fixed area. Let’s make all
these ideas more precise, for any kind of distribution.
3.1.3 Mean and variance
Theaverage(ormeanorexpectation value)ofxfor any distribution is written〈x〉and defined
by
〈x〉={∑
∫ ixiP(xi),discrete
dxxP(x),continuous.(3.9)
Forthe uniform and Gaussian distributions, the mean is just the center point. That’s because these
distributions are symmetrical: There are exactly as many observations a distancedto the right of
the center as there are a distancedto the left of center. For a more complicated distribution this
needn’t be true; moreover, the mean may not be the same as themost probable value,which is
the place whereP(x)ismaximum.
More generally, even if we know the distribution ofxwemay instead want the mean value of
some other quantityf(x)depending onx.Wecan find〈f〉via
〈f〉={∑
∫if(xi)P(xi),discrete
dxf(x)P(x),continuous.(3.10)
If you go out and measurexjust once you won’t necessarily get〈x〉right on the nose. There
is some spread, which we measure using theroot-mean-square deviation(orRMS deviation,or
standard deviation):
RMS deviation =
√
〈(x−〈x〉)^2 〉. (3.11)Example a. Show that〈(〈f〉)〉=〈f〉for any functionfofx.
b. Show that if the RMS deviation equals zero, this implies that every measurement
ofxreally does give exactly〈x〉.
Solution:
a. We just note that〈f〉is a constant (that is, a number), independent ofx.The
average of a constant is just that constant.
b. In the formula 0 =〈(x−〈x〉)^2 〉=∑
iP(xi)(xi−〈x〉)(^2) ,the right hand side doesn’t
have any negative terms. The only way this sum could equal zero is for every term
to be zero separately, which in turn requires thatP(xi)= 0 unlessxi=〈x〉.
Note that it’s crucial to square the quantity (x−〈x〉)when defining the RMS deviation; otherwise
we’d trivially get zero for the average value〈(x−〈x〉)〉.Then we take the square root just to get
something with the same dimensions asx.We’ll refer to〈(x−〈x〉)^2 〉as thevarianceofx(some
authors call it thesecond momentofP(x)).
Your Turn 3b
a. Show that variance(x)=〈x^2 〉−(〈x〉)^2.
b. Show for the uniform distribution (Equation 3.5) that variance(x)=a^2 /12.