By Steve Dobbs, Jane Miller, Julian Gilbey
Written to compare the contents of the Cambridge syllabus. records 2 corresponds to unit S2. It covers the Poisson distribution, linear mixtures of random variables, non-stop random variables, sampling and estimation, and speculation exams.
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Extra resources for Advanced Level Mathematics: Statistics 2
L! l¼1 k¼1 n X 1 k X l l k¼1 l¼1 ð2:174Þ Thus, the cumulant of an arbitrary order of the sum can be expressed as a weighted sum of the cumulants of the same order. In particular, eq. 174), written for the average m and the variance D ¼ 2 , produces the following rule m ¼ n X ak mk ; k¼1 D ¼ n X a2k Dk ð2:175Þ k¼1 If all variables have the same distribution and all the weights are equal, ai ¼ a, then eq. 172) is reduced to Â ð j uÞ ¼ Ân ð j u aÞ; É ð j uÞ ¼ n Éð j u aÞ; k ¼ n ak k ð2:176Þ Since for an arbitrary Gaussian random variable all the cumulants of order higher than two are equal to zero, eq.
1 þ2 þÁÁÁþn 1 1 1 ln Ân ðu1 ; u2 ; . . ; un Þ @ u1 @ u1 Á Á Á @ u1 u1 ¼u2 ¼ÁÁÁ¼un ¼0 ð2:126Þ A relation between moments and cumulants of a random vector can be established in a way similar to relations obtained for moments and cumulants of a random variable. Finally, conditional densities can be used to deﬁne conditional moments and cumulants. 31 RANDOM VECTORS AND THEIR DESCRIPTION As in the case of a random variable, lower order moments and cumulants play a prominent role in the description of a random vector.
Xn Þ ð2:117Þ In particular, the following formula (playing a prominent role in the theory of Markov processes) can be obtained ð1 pðx1 jx2 ; x3 Þpðx2 jx3 Þd x2 ð2:118Þ pðx1 jx3 Þ ¼ À1 All the considered deﬁnitions and rules remain valid for the case of a discrete random variable, with integrals being reduced to sums. Random variables, 1 ; 2 ; . . ; n are called mutually independent if events f1 < x1 g; f2 < x2 g; . . ; fn < xn g are independent for any values of xi ; i ¼ 1; . . ; n. e.