Mathematical Methods for Physicists: A concise introduction - Site Map

INTRODUCTION TO PROBABILITY THEORY
distribution or probability function for the random variable X. Evidently any
probability distribution p i for a discrete random variable must satisfy the following
conditions:
(i) 0 p i 1;
(ii) the sum of all the probabilities must be unity (certainty), P i p i ˆ 1:
Expectation and variance
The expectation or expected value or mean of a random variable is de®ned in
terms of a weighted average of outcomes, where the weighting is equal to the
probability p i with which x i occurs. That is, if X is a random variable that can take
the values x 1 ; x 2 ; ...; with probabilities p 1 ; p 2 ; ...; then the expectation or
expected value E…X† is de®ned by
E…X† ˆp 1 x 1 ‡ p 2 x 2 ‡ˆX
p i x i :
…14:14†
Some authors prefer to use the symbol for the expectation value E…X†. For the
three dimes tossed at the same time, we have
and
x i ˆ 0 1 2 3
p i ˆ 1=8 3=8 3=8 1=8
E…X† ˆ1
8 0 ‡ 3 8 1 ‡ 3 8 2 ‡ 1 8 3 ˆ 3
2 :
We often want to know how much the individual outcomes are scattered away
from the mean. A quantity measure of the spread is the di€erence X E…X† and
this is called the deviation or residual. But the expectation value of the deviations
is always zero:
E…X E…X†† ˆ X
…x i E…X††p i ˆ X
x i p i E…X† X p i
i
i
i
ˆ E…X†E…X†1 ˆ 0:
This should not be particularly surprising; some of the deviations are positive, and
some are negative, and so the mean of the deviations is zero. This means that the
mean of the deviations is not very useful as a measure of spread. We get around
the problem of handling the negative deviations by squaring each deviation,
thereby obtaining a quantity that is always positive. Its expectation value is called
the variance of the set of observations and is denoted by 2
i
2 ˆ E‰…X E…X†† 2 ŠˆE‰…X † 2 Š:
…14:15†
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