What is Generating function: Definition and 127 Discussions
In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.
There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.
Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations defined for formal series. These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of x. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal series as its series expansion; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a convergent series when a nonzero numeric value is substituted for x. Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series; for example, negative and fractional powers of x are examples of functions that do not have a corresponding formal power series.
Generating functions are not functions in the formal sense of a mapping from a domain to a codomain. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence of term coefficients.
Hi.
I'm really struggling with this generating function problem. Any help would be greatly appreciated.
Question:
Find the generating function for the compositions (c1,c2,c3...,ck) such that for each i, ci is an odd integer at least 2i-1.
Second part of question:
Use the above...
Homework Statement
Let Y1,Y2,... be independent identically distributed random variables with
P(Y1=1) = P(Y1=-1) = 1/2 and set X0 = 1, Xn=X0 + Y1 +...+ Yn for n >= 1. Define
H0 = inf { n >= 0 : Xn = 0}.
Find the probability generating function Φ(s) = E(sH0).
Homework Equations...
Homework Statement
Suppose \theta ~ ~ gamma(\alpha , \lambda) where alpha is a positive integer. Conditional on \theta, X has a Poission distribution with mean \theta . Find the unconditional distribution of X by finding it's MGT.
Homework Equations
The Attempt at a Solution...
Homework Statement
Let X be uniformly distributed over the unit interval (0,1). Determine the moment generating function of X, and using this, determine all moments around the origin.
Homework Equations
The Attempt at a Solution
I know that the MGT is M(x) = E[ext]
I'm just...
Homework Statement
Let X denote a random variable with the following probability mass function:
P(j)= 2^(-j), j=1,2,3,...
(a) Compute the moment generating function of X.
(b) Use your answer to part (a) to compute the expectation of X.
Homework Equations
m.g.f of X is M (t) =...
J.J. Sakurai Modern Quantum Mechanics p. 74
It says,
[A,H] = 0;
H|a'> = Ea' |a'>
where H is the Hamiltonian A is any observable |a'> is eigenket of A
then,
exp ( -iHt/h)|a'> = exp (-iEa't/h)|a'>
where h is the reduced Planck's constant.
I want to know WHY ?
and besides, I would...
Homework Statement
Consider a branching process with branching probabilities given by P0=1/2 and Pj=1/3^{j} for j \geq 1
Find the probability generating function: \sum^{\infty}_{n=0} p_{n}x^{n}
The Attempt at a Solution
Now, the answer is supposed to be G(x) = (x+3)/(2(3-x)), but I...
Homework Statement
The problem asks to use the Lagrange form of the remainder in Taylor's Theorem to prove that the Maclaurin series generated by f(x) = xex converges to f. From the actual answer, I'm guessing it wants me to use the Remainder Estimation Theorem to accomplish this...
Homework Statement
Suppose RX(t) = E[(1 − tX)−1] is called the geometric generating function
of X. Suppose the random variable Y has a uniform distribution on (0, 1); ie
fY (y) = 1 for 0 < y < 1. Determine the geometric generating function of Y .
Homework Equations
The Attempt at...
Homework Statement
Let f(x) = 2x 0<x<1
a) Determing the Moment Generating function M(t) of X
b) Use the MGT to determine all moments about the origin
c) Give the 3rd central moment called the skewness
Homework Equations
The Attempt at a Solution
a) \int^1_0 e^{tx}2x dx =...
Homework Statement
Find the mgf of 2/25*(5-y) fo 0<y<5
Homework Equations
M(t) = INT e^yt f(y)dy
The Attempt at a Solution
= (2*(e^(5t)-5t-1))/25t
Is this ok
1. Let X denote the mean of a random sample of size 75 from the distribution that has the pdf f (x) =1, 0<=x<=1. Calculate P (0.45 <X< 0.55).
2. Derive the moment-generating function for the normal density.
3. Let Y n (or Y for simplicity) be b (n, p). Thus, Y / n is approximately N [p, p (1...
The Legendre functions may be defined in terms of a generating function: g(x,t) = \frac{1}{\sqrt{1-2xt+t^2}}
Of course, \frac{1}{\sqrt{1+x}} =\sum^{\infty}_{n=0} (\stackrel{-.5}{n})x^n .
However, this series doesn't converge for all x. It only converges if |x| < 1. In our case, |t^2 -...
Homework Statement
The Bessel function generating function is
e^{\frac{t}{2}(z-\frac{1}{z})} = \sum_{n=-\infty}^\infty J_n(t)z^n
Show
J_n(t) = \frac{1}{\pi} \int_0^\pi cos(tsin(\vartheta)-n\vartheta)d\vartheta
Homework Equations
The Attempt at a Solution
So far I...
Homework Statement
Use the Legendre generating function to show that for A > 1,
\int^{\pi}_{0} \frac{\left(Acos\theta + 1\right)sin\thetad\theta}{\left(A^{2}+2Acos\theta+1\right)^{1/2}} = \frac{4}{3A}
Homework Equations
The Legendre generating function
\phi\left(-cos\theta,A\right) =...
Homework Statement
Use conditional expectation to compute the moment generating function M_z(s) of the random variable Z=XT.
Homework Equations
X ~ R(0,10)
T ~ exp(0.1)
The Attempt at a Solution
By definition:
M_z(s) = E(exp(sZ))
=E(exp(sXT))
The only thing I can think...
Hi,
I don't understand, in general, how am I supposed to find an appropriate generating function to a given canonical transformation. It seems to me like a lot of guesswork. Can anyone give me some guidelines?
thanks.
Homework Statement
Find a linear homogeneous recurrence relation satisfied by an=2^n+n!
Homework Equations
The Attempt at a Solution
The teacher gave us a hint using generating functions. The generating function for f(x) is
f(x)=1+2x+4x^2+8x^3+...+1+x+2x^2+6x^3+24x^4+... The...
I'm given the probability density function:
f(x) = 3x^2 for x in [0, 1]
f(x) = 0 elsewhere
I want to find E[X^2] which is easy if I use the integral definition (I got 3/5). Yet, when I try and do this using Moment Generating Function (mgf) I cannot seem to get the same answer (in...
A probability distribution,f(x) ,can be represented as a generating function,G(n) , as \sum_{x} f(x) n^x . The expectation of f(x) can be got from G'(1) .
A bivariate generating function, G(m,n) of the joint distribution f(x,y) can be represented as \sum_{x} \sum_{y} f(x,y) n^x m^y ...
Given that the moment generating function of a random variable is
(e^t)/(2-e^t) is there a way I can go backwards and find the pdf, or could 2 different pdf's have the same mgf?
i have X_1,X_2,...X_n independant poisson-distributed variables with parameters: alfa_i and i=1,...k(unsure about this. however says so in the excercise)
i am supposed to find the distribution of
Y= SUM(from 1 to n) a_i*X_i where a_i>0
maybe one could use the "poisson paradigm" by...
I'm sure this is relatively easy, but after an hour or so googling, I can't seem to find the formula for generating terms of the http://steiner.math.nthu.edu.tw/chuan/123/test/euler.htm
Is this known by some other name? Maybe that's why I can't find it?
Thanks
I have absolutly no idea how to do this.
so let X be a random variable with pdf fx(xy) =
x for 0<=x<=1
2 - x for 1 <= 1 <= 2
0 otherwise.
I"m looking through my book, and it doesn't give examples that resembles this.
all I see is the moment is e^(tk) * the function...
and tI...
Let be a series Sum(0,infinite)anX^n=f(x) then could it be inverted to get the an?..in fact if we set x=1/Z the series becomes
anZ^-n=Zeta transform of an so we could invert this (as seen in mathworld.com),could it be done to get the an?..thanks.
Q: show that (1-4x)^(-1/2) generates the sequence 2n chooses n, n is defined as natural
All the formulas I have requires integer exponent. I am not sure how to deal with (-1/2).
Thanks for any input!