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!