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Unformatted text preview: Time Series Analysis Essential Concepts G. P. Nason 1st October 1998 (updated 28th September 2005) Essential concepts for time series analysis This chapter reiterates important concepts that you should be aware of from the first year. Section 0.9 is a small and obvious bit from Statistics 2. 0.1 Random variables Let X be a random variable (r.v.) taking values in R . The distribution function F of X is given by F ( x ) = P ( X x ) (1) where P ( A ) is the probability of event A occurring. If X is a continuous r.v. its distribution function F may be written as F ( x ) = Z x- f ( u ) du (2) for some x R and integrable function f : R [0 , ). The function f is called the density function of X . 0.2 Properties of density functions A density function f always integrates to 1. In mathematical notation: Z - f ( x ) dx = 1 . (3) 0.3 Expectation Define the expectation or mean of X as E X = Z - xf ( x ) dx. (4) Note that the expectation can be equal to and in this case we say that the moment does not exist (the mean is the first moment). Expectation is linear . That is, if a, b R are constants and X , Y are r.v.s then E ( aX + bY ) = a E X + b E Y. (5) 1 0.4 Variance The variance of a r.v. X is defined to be var( X ) = E ( X- E X ) 2 (6) and describes the variation of...
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