(54) |

(55) |

In this situation, it is useful to regard the probability as a smooth function of . Let be a continuous variable which 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

(57) |

(58) |

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

(60) |

(61) |

(62) |

(63) | |||

(64) |

if corresponds to the

It follows from Eq. (59) that

(66) |

(67) |

valid for , is called

According to Eq. (65),

(69) |

(70) |

(71) |

Further differentiation of Eq. (65) yields

(72) |

(73) |

It follows from the above that the Taylor expansion of can be written

(74) |

(75) |

(76) |

(77) |

As is well-known,

(80) |

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 Eq. (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* centred on
. Hence, this curve is sometimes
called a *bell curve*.
At one standard deviation away from the mean value, *i.e.*,
, 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 indeed 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
. This is a very well-known result.

In the above analysis, we have gone from a *discrete* probability
function to a *continuous* probability density .
The normalization condition becomes

(83) |

respectively. These results follow as simple generalizations of previously established results for the discrete function . The limits of integration in the above expressions can be approximated as because is only non-negligible in a relatively narrow range of . Finally, it is easily demonstrated that Eqs. (82)-(84) are indeed true by substituting in the Gaussian probability density, Eq. (81), and then performing a few elementary integrals.