Answered: Conditioned branching process. Consider… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 67E

See similar textbooks

Related questions

Question

Conditioned branching process. Consider a branching process whose family sizes have the
geometric mass function f(k): apk, k≥ 0, where μ = p/q > 1. Let Zn be the size of the nth
generation, and assume Zo = 1. Show that the conditional distribution of Zn/u", given that Zn > - 0,
converges as n → ∞ to the exponential distribution with parameter 1 - µ-¹.
=

Expert Solution

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Blurred answer

Similar questions

Let Y1.,Yn be random sample from a Gamma (a, ß), show 1/nE, Y? converges in probability %3D1 and find its limit.
The X be a continuous random variable with cumulative distribution function 1 F(x) = 1 for x > 0 (and zero otherwise). - 1+ x Find the cumulative distribution function of Y = log X and sketch its curve.
iid 1. Let X1, X2,..., X, * Bernoulli(1/2). For n = 1,2,..., define Yn = E-1 Xi/2'. (a) Find the pdf/pmf for Yn. (b) Investigate the convergence in distribution of Yn.
Let X1, X2, ..., Xn be a sequence of independent and identically distributedrandom variables having the Exponential(λ) distribution, λ > 0,fXi(x) = λe−λx , x > 00 , otherwise(a) Show that the moment generating function mX(s) := E(e^sX) = λ/λ−s for s < λ;(b) Using (a) find the expected value E(Xi) and the variance Var(Xi).(c) Define the random variable Y = X1 + X2 +· · ·+ Xn. Find E(Y ), Var(Y ) and the moment generating function of Y .(d) Consider a random variable X having Gamma(α, λ) distribution,fX(x) = (λαxα-1/Γ(α)) e−λx , x > 00 , otherwiseShow that the moment generating function of the random variable X is mX(s) =λα 1/(λ−s)α for s < λ, where Γ(α) isΓ(α) = (integral from 0 to inifity ) xα−1e−xdx.(e) What is the probability distribution of Y given in (c)? Explain youranswer.
d) Let Y be a continuous random variable with a probability mass function: - 1 fV) = (,) p"(1– p)"-* - 1 Derive the moment generating function (MGF) of a random variable Y.
Suppose Xn is the sample mean of a random sample of size n from a distribution that has a Gamma(2,1/2) (where a = 2, B = }) pdf f(x) = 4xe-2a, 0 < x < ∞, zero elsewhere. Use the Central Limit Theorem to deduce that the random variables Vn(Xn – 1) converges in distribution to N(0, 1/2).
(a) Let {Y} be a stationary ARMA(1, 1) model with mean 0 and white noise sequence {et} ~ WN(0,0²). Calculate E(etYt+k), for all k ≥ 0 and express your answer in terms of Kronecker-Delta functions where possible.
On what interval(s) of P is the population increasing? dP P 1.5P (1- dt 5000/
A continuous random variable T (in years) of the lifespan of a device has the following probability density function f (t) = {3 pt", Osts2 otherwise where p is a constant, (a) Determine the value of p. (b) Find the cumulative distribution function of T, F(t). (c) If U =ÉT+1 determine E(U) and Var(U).
Experiment X is consist of drawing two balls from a basket containing 13 blue and 5 red balls. Experiment Y in extracting one more ball from the same basket. What the entropy H(y) of experience Y and the conditional entropy H(Y/X) of this experience?
Let R have probability mass function (pmf) p(r)=1/8 for r=1,2,...,8. Find (1)the cumulative distribution function (cdf) of R; (2)P(R>5); (3)E(R); (4)E(R²): (5)E[(R-3)(R-1)]; (6) Var(R).
s) Let X be a discrete random variable taking values {x1, x2, . . . , xn} with probability {p1, p2, . . . , pn}. The entropyof the random variable is defined asH(X) = − Σn i=1 pi*log(pi)Find the probability mass function for the above discrete random variable that maximizes the entropy.

SEE MORE QUESTIONS

Recommended textbooks for you

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS