using the generating function "gm(x)" for Stirling numbers of the first kind
Using the generating function "gm(x)" for Stirling numbers of the first kind
find a closed formula for a(x)=(summation k=0 to x) S(k,3) S(k,3) are stirling numbers of 2nd kind
Q. 5. Let X be any random variable, with moment generating function M(S) = E[es], and assume M(s) < o for all s E R. The cumulant generating function of X is defined as A(s) = log Ele**] = log M(s), SER Show the following identities: (1) A'(0) = E[X]. (2) A”(0) = Var(X). (3) A"(0) = E[(X - E[X]))). Using the inversion theorem for MGFs, argue the following: (4) If A'(s) = 0 for all s ER, then P(X=...
3. (a) Consider two vectors x and x' with x >x'). Use the generating function for the Legendre polynomial and the addition theorem for spherical harmonics to show: - 47 x'' 1 x - x' 2221 + 1 Yom (0.6) I=0 m=-, 21 +1xl+"Im(0, (b) Use this formula to compute the potential at point x due to a line of charge of uniform density extending on the y-axis from y = 0 to y = L, where L < x.
If two random variables have the same generating function, must they have the same cumulative distribution function? L.8) Central Limit Theorem One version of Central Limit Theorem says this: Go with independent random variables (Xi, X2, X3, ..., X.....] all with the same cumulative distribution function so that: 11-Expect[Xi]-Expect[s] and σ. varpk-VarX] for all i and j . Put: s[n] = As n gets large, the cumulative distribution function of S[n] is well approximated by the Normal[o, 1] cumulative distribution...
Solve and show work for problem 8 Problem 8. Consider the sequence defined by ao = 1, ai-3, and a',--2an-i-an-2 for n Use the generating function for this sequence to find an explicit (closed) formula for a 2. Problem 1. Let n 2 k. Prove that there are ktS(n, k) surjective functions (n]lk Problem 2. Let n 2 3. Find and prove an explicit formula for the Stirling numbers of the second kind S(n, n-2). Problem 3. Let n 2...
1. Let X have probability generating function Gx (s) and let un generating function U(s) of the sequence uo, u1, ... satisfies P(X > n). Show that the (1- s)U(s) = 1 - Gx(s), whenever the series defining these generating functions converge. 1. Let X have probability generating function Gx (s) and let un generating function U(s) of the sequence uo, u1, ... satisfies P(X > n). Show that the (1- s)U(s) = 1 - Gx(s), whenever the series defining...
7. Derive the moment-generating function M(t) for X 1(a, X). 8. Expand the moment-generating function M(t) = ex+oft®/2 in a power series in t to compute E[X3] if X ~ N(1, 2).
3. Use the probability generating function Px)(s) to find (a) E[X(10)] (b) VarX(10)] (c) P(X(5)-2) . ( 4.2 Probability Generating Functions The probability generating function (PGF) is a useful tool for dealing with discrete random variables taking values 0,1, 2, Its particular strength is that it gives us an easy way of characterizing the distribution of X +Y when X and Y are independent In general it is difficult to find the distribution of a sum using the traditional probability...
a) Find the moment generating function (mgf) of X. b) Using part a), that is the mgf of X, find the expected value (E[X]) and (V ar[X]). Let X be a random variable such that -1, with probability q 1 with probability 1-q,
7. State Taylor's theorem for a function f(x, y) of two variables and prove it by using Taylor's theorem for a single variable function. 7. State Taylor's theorem for a function f(x, y) of two variables and prove it by using Taylor's theorem for a single variable function.