4 - MEANS AND VARIANCES The term expectation (or mean) has so far been confined to a single random variable. Related to expectation is the term variance and this will be discussed before two important new distributions are introduced. Expectation and variance will then be applied to cases where there are two or more random variables. Derived Random Variables This course follows the common practice of using X and r to refer to the name and value of a single random variable. The most-frequently used example of a random variable has been the outcome of throwing a die. If two people are betting on the outcome of throwing a die and wish to make matters (ever so slightly) more interesting they might decide to bet on some function of the outcome instead of the outcome itself. They might perhaps bet on the square of the outcome, the sine of the outcome, seven more than the outcome, two to the power of the outcome and so on. Such a decision does not affect the number of possible outcomes or the values of the outcomes themselves. What is affected is the values the people bet on and, of much greater interest, the long-term average of the values bet on. Any function of some random variable X is itself a random variable; call it Y where: Y = f(X) Subject to certain common-sense constraints f is an arbitrary function and Y is called a derived random variable. If f is like the square function it will lead to integer values. If it is like the sine function it will not and various issues arise which will not be addressed here. Generalised Expectation Fortunately the expectation of some function f(X) of a random variable X is a trivial generalisation of (3.1): E (4.1) r This can be derived in exactly the same way as the simple expectation by considering the position of the centre of gravity of a light beam supporting weights. Suppose the bets are on the square of the outcome of throwing a die. The weights are just the same but they are placed at positions 0, 1, 4, 9, 16, 25 and 36 along the beam. The analysis gives rise to the expectation: E(X2) = 0 .- + 1 .- + 4 .- + 9 .- + 16. - + 25. - + 36." 1 6 1 6 91 - = 15- Many functions of random variables will be used in this course but the square function and one closely related to the square are undoubtedly the most important. - 4.1 -
Variance Suppose that, instead of simply squaring the outcome of each throw, the players first subtract the mean (3- of course) and then square. The expectation of the values bet on now is: E(X - 32)2) = > (r - 3})2.P (X = r) r This value is known as the variance (usually denoted by the letter V) or mean squared deviation from the mean usually denoted by 2.