Thus, making use of Equation (2.27), the mean number of occurrences of outcome 1 in observations is given by

(2.39) |

We can see that if the final factor were absent on the right-hand side of the previous expression then it would just reduce to the binomial expansion, which we know how to sum. [See Equation (2.23).] We can take advantage of this fact using a rather elegant mathematical sleight of hand. Observe that because

(2.40) |

the previous summation can be rewritten as

(2.41) |

The term in square brackets is now the familiar binomial expansion, and can be written more succinctly as . Thus,

(2.42) |

However, for the case in hand [see Equation (2.11)], so

In fact, we could have guessed the previous result. By definition, the probability, , is the number of occurrences of the outcome 1 divided by the number of trials, in the limit as the number of trials goes to infinity:

(2.44) |

If we think carefully, however, we can see that taking the limit as the number of trials goes to infinity is equivalent to taking the mean value, so that

(2.45) |

But, this is just a simple rearrangement of Equation (2.43).

Let us now calculate the variance of . Recall, from Equation (2.36), that

(2.46) |

We already know , so we just need to calculate . This average is written

(2.47) |

The sum can be evaluated using a simple extension of the mathematical trick that we used previously to evaluate . Because

(2.48) |

then

(2.49) |

Using , we obtain

(2.50) |

because . [See Equation (2.43).] It follows that the variance of is given by

(2.51) |

The standard deviation of is the square root of the variance [see Equation (2.37)], so that

(2.52) |

Recall that this quantity is essentially the width of the range over which is distributed around its mean value. The relative width of the distribution is characterized by

(2.53) |

It is clear, from this formula, that the relative width decreases with increasing like . So, the greater the number of trials, the more likely it is that an observation of will yield a result that is relatively close to the mean value, .