Gaussian Probability Distribution

(2.54) |

In this limit, the standard deviation of is also much greater than unity,

implying that there are very many probable values of scattered about the mean value, . This suggests that the probability of obtaining occurrences of outcome 1 does not change significantly in going from one possible value of to an adjacent value. In other words,

In this situation, it is useful to regard the probability as a smooth function of . Let be a continuous variable that is interpreted as the number of occurrences of outcome 1 (after observations) whenever it takes on a positive integer value. The probability that lies between and is defined

(2.57) |

where is called the

(2.58) |

which is equivalent to smearing out the discrete probability over the range . Given Equations (2.38) and (2.56), the previous relation can be approximated as

For large , the relative width of the probability distribution function is small: that is,

(2.60) |

This suggests that is strongly peaked around the mean value, . Suppose that attains its maximum value at (where we expect ). Let us Taylor expand around . Note that we are expanding the slowly-varying function , rather than the rapidly-varying function , because the Taylor expansion of does not converge sufficiently rapidly in the vicinity of to be useful. We can write

(2.61) |

where

By definition,

(2.63) | ||

(2.64) |

if corresponds to the maximum value of .

It follows from Equation (2.59) that

If is a large integer, such that , then is almost a continuous function of , because changes by only a relatively small amount when is incremented by unity. Hence,

(2.66) |

giving

for . The integral of this relation

valid for , is called

According to Equations (2.62), (2.65), and (2.67),

Hence, if then

(2.70) |

giving

(2.71) |

because . [See Equations (2.11) and (2.43).] Thus, the maximum of occurs exactly at the mean value of , which equals .

Further differentiation of Equation (2.69) yields [see Equation (2.62)]

(2.72) |

because . Note that , as required. According to Equation (2.55), the previous relation can also be written

(2.73) |

It follows, from the previous analysis, that the Taylor expansion of can be written

(2.74) |

Taking the exponential of both sides, we obtain

(2.75) |

The constant is most conveniently fixed by making use of the normalization condition

(2.76) |

which becomes

(2.77) |

for a continuous distribution function. Because we only expect to be significant when lies in the relatively narrow range , the limits of integration in the previous expression can be replaced by with negligible error. Thus,

As is well known,

(See Exercise 1.) It follows from the normalization condition (2.78) that

(2.80) |

Finally, we obtain

This is the famous

Suppose we were to
plot the probability
against the integer variable
, and then
fit a continuous curve through the discrete points thus obtained. This curve
would be
equivalent to the continuous probability density curve
, where
is the continuous version of
. According to Equation (2.81), the
probability density attains its maximum
value when
equals the mean
of
, and
is also symmetric about this point. In fact, when plotted with the
appropriate ratio of vertical to horizontal scalings, the Gaussian probability
density curve looks rather like the outline of a
bell centered on
. Hence, this curve is sometimes
called a *bell curve*.
At one standard deviation away from the mean value--that is
--the probability density is
about 61% of its peak value. At two standard deviations away from the mean
value, the probability density is about 13.5% of its peak value.
Finally,
at three standard deviations away from the mean value, the probability
density is only about 1% of its peak value. We conclude
that there is
very little chance that
lies more than about three standard deviations
away from its mean value. In other words,
is almost certain to lie in the
relatively narrow range
.

In the previous analysis, we went from a discrete probability function, , to a continuous probability density, . The normalization condition becomes

under this transformation. Likewise, the evaluations of the mean and variance of the distribution are written

(2.83) |

and

respectively. These results follow as simple generalizations of previously established results for the discrete function . The limits of integration in the previous expressions can be approximated as because is only non-negligible in a relatively narrow range of . Finally, it is easily demonstrated that Equations (2.82)-(2.84) are indeed true by substituting in the Gaussian probability density, Equation (2.81), and then performing a few elementary integrals. (See Exercise 3.)