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30 Aug 2019: State the dual Cauchy identity. Let F (t) =. j>0 fj tj be a formal power series, where f0 = 1.

30 Aug 2019: Xj = min(Tj , cj ), for 1 j n. extending the notation of (b) in the natural way.

30 Aug 2019: Theeffects of recombination in the segment may be ignored. The time for which the samplehas j distinct ancestors is denoted by Tj , j = 2, 3,. ,

17 Jun 2019: Find the probability-generating functions of the random variables Hj and Tj.

30 Aug 2019: Let uji be the approximation at xi and tj ,i = 1,. ,

4 Aug 2022: Denote the kernel valuesfor indexed times as Rij k(ti, tj) and let R k(0,0).

30 Aug 2019: Show that the action of the left-invariant vector fields,. Tj (u) = iµji(u). ... ui,. on the matrix A is given by Tj A(u) = A(u) σj , and hence that Ti are generators of the.

17 Jun 2019: for j 6= i. Prove that the random vectors Yj AjX are independent, and thatY (Y T1 ,. ,Y TJ )T has a multivariate normal distribution.[ Hint: Random vectors are independent if

17 Jun 2019: Assume p > q. Let Tj = inf{n > 1 : Xn = j} if this is finite, and Tj = otherwise. ... Let Ti = inf{n > 1 : Xn = i}. For each i 6= j letqij = P(Tj < Ti | X0 = i) and mij = E(Tj | X0 = i).

30 Aug 2019: tJ(t, x, v) = (V F(t, X, V )) J(t, x, v),.