(a)
(b) = =
(c) =
(d) =
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2.1. Find the ordinary generating functions for the following sequences. (a) (1, 1, 1, 1, 0,...
Section 1.7: 4. Let f(x) be the exponential generating funcion of a sequence {%). Find the exponential generating functions for the follow- ing sequences in terms of f(x): (a) fan cl (b) foan (c (nani (e) 0, a,a, , (g) ao,0, a2,0, a,0,... (h) a, a2, a,... 8. (a) A sequence a satisfies the recurrence relation a3an+2, ao0 Find the exponential generating function ΣΧ0Lnz" Section 1.7: 4. Let f(x) be the exponential generating funcion of a sequence {%). Find the...
Find a closed form for the generating function for each of the following sequences: an=5-2n 0, 0, 0, 4, -12, 36, -108 0, 1, 0, 4, 0, 16, 0, 64, 0, …
11) Find a closed form for the generating function for each of the following sequences: a) ?? = 5(−2)? g) 0, 0, 0, 4, -12, 36, -108 h) 0, 1, 0, 4, 0, 16, 0, 64, 0, ...
) Find a closed form for the generating function for each of the following sequences: an=5-2n 0, 0, 0, 4, -12, 36, -108 0, 1, 0, 4, 0, 16, 0, 64, 0, …
5*. Consider all sequences (ai,. .., an) such that a, are nonnegative integers and a ai+ 2. Let P, n and Rn be the number of such sequences which start from 0, 1 and 2 respectively. (a) Compute P, Qn, Rn by writing down all such sequences for n 1,2,3. (b) Prove that P, Qn Rn satisfy the recurrence relations: (c) Translate the above equations into linear equations for the generating functions for P, Qn, Rn (d) Solve these equations...
4 S and Gy(s) Consider the following two probability generating functions Tx(s) 1-2s 1+3s respectively for the random variables Xand Y a) Find the expectations of the random variables X and Y b) Find the probability that X-0; c) Find the probability that Y 0; d) Find the probability that X+Y 0;
L.1) Generating functions and discrete random variables a) The data set is X-0, 1, 2, 2, 3, 3, 3 What is et* ? b) The data set is X-0, 1, 2, 2, 3, 3, 3) Give a formula for the generating function of X. c) How is the generating function of X related to ExpectTe]? L.2) Generating functions and discrete random variables a) The random variable is a pull from (0, 1, 2, 2, 3, 3, 3 Give a formula...
Use generating functions to solve the following recurrence. T(0) = 0, T(1) = 1. T(n) = 7 T(n-1) – 12 T(n-2)
Assume that the generating function of the nonnegative integer random variable ξ is G(S) Find the generating functions for the following sequences (1) an=P{ξ≤n} (2) bn=P{ξ=2n}
Using generating functions, find the number of solutions of the equation (For ф type C(6,4), and for 5l type fact (5).) Using generating functions, find the number of solutions of the equation 7. (For φ type C(6,4).) Using generating functions, find the number of solutions of the equation (For φ type C(6,4).) Using generating functions, find the number of solutions of the equation (For ф type C(6,4), and for 5l type fact (5).) Using generating functions, find the number of...