et X1, X2, X3, . . . be a sequence of independent Poisson distributed random variables with parameter 1. For n ≥ 1 let Sn = X1 + · · · + Xn. (a) Show that GXi(s) = es−1. (b) Deduce from part (a) that GSn(s) = ens−n

et X1, X2, X3, . . . be a sequence of independent Poisson distributed random variables with parameter 1. For n ≥ 1 let Sn = X1 + · · · + Xn. (a) Show that GXi(s) = es−1. (b) Deduce from part (a) that GSn(s) = ens−n

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Related questions

Q: (6) Let X be a Poisson random variable with parameter 0, then E(X²) = 1. От OF

A: Poisson distribution is the type of discrete probability distribution. It explain the number of…

Q: Let X be a random variable such that E(X²) a. V(3x-3) b. E(3x-3) = 58 and V(X) = 40. Compute

Q: Assume we have a linear model with 2 X variables, X1, X2. Show that the variance inflation factors…

A: variance inflation factor (VIF) : is the ratio of the variance of estimating some parameter in a…

Q: Suppose that X₁, the number of particles emitted in t hours from a radio - active source has a…

Q: S Suppose that the time (in hours) required to repair a machine is an exponentially distributed…

A: The distribution of the repairing time for the machine is specified to be distributed exponentially…

Q: 5. Let Y,, Y2, ., Yn be independent, exponentially distributed random variables with mean 0/2. Show…

A: Solution

Q: Let X be a random variable with p.d.f. 4k 1< x<10 f (x) =< find k. O.W

Q: Suppose the time it takes Alex to do this exam is exponentially distributed with parameter 3 per…

A: Introduction to Exponential distribution: A random variable X is said to have a exponential…

Q: d) The random variable X has a Poisson distribution such that P(X= 0) = P(X = 1). Calculate P(X =…

A: Given problem Given that d) The random variable X has a Poisson distribution such that P( X=0…

Q: Let X be a uniform random variable over the interval 1 to 4 and Y is exponential with a mean of 2.…

A: Given information: X~Uni(a=1,b=4) Standard deviation of X is: σX=b-a12 =4-112 =0.8660…

Q: 2. Let X be a random variable with the Poisson distribution, parameter 2. Show that, for w = 1, 2,…

A: X~Poisson(λ) Therefore P(X=x)=e-λλxx! ; x=0,1,2,……

Q: Let M and N be two independent events such that P(M) = 1/3 and P(N) = 1/4. Find P(M/N')

Q: Suppose §1, $2, ... are iid random variables having mean u and variance o?. Form the random sum SN =…

A: Given that ξ1, ξ2, ..., ξn are iid random variables having mean μ and variance σ2. SN=ξ1+ξ2+...+ξN

Q: Suppose that E and F are two events such that Pr[E] = 17/29, Pr[F] = 13/29, and Pr[EN F] = 8/29.…

A: Given, P(E) = 1729P(F) = 1329P(E∩F)=829

Q: (6) Let X be a Poisson random variable with parameter 9, then E(X²) = 1.

A: Given X~Pois(θ)

Q: (b) Determine e so that п п c£[(x; - X;)² i=l j=1 is an unbiased estimator of o². s2 ogunl to

A: 6. From the given information, X1,.....,Xn~f be an iid random sample with mean μ and variance σ2. a)…

Q: Suppose the random variable y is a function of several independent random variables, say…

A: Given that the random variable y is a function of several independent random variables, say…

Q: Suppose that Y₁ = 0.5, Y₂ = 0.6, Y3 = 0.2, Y4 = 0.7 and Y5 = 0.6, represents a random sample. Each…

A: Solution

Q: Let Y₁, Y2,..., Yn denote a random sample of size n from a population with a uniform distribution on…

Q: Page 327, 5.1.7.* Let X₁, ...,Xn be iid random variables with common pdf f(x) exp{-(x-20)}, x > 20,-…

Q: Let Xi and Yi be random variables with Var(Xi) = σx2 and Var(Yi) = σy2 for all i ∈ {1, . . . , n}.…

A: It is given that: varxi=σx2varyi=σy2 and xi and yi are independent and corrxi,yi=ρ

Q: Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform…

A: Let X1, X2,.......Xn be the random sample having Uniform distribution (0,1)Y1=min{X1,…

Q: Let X, X,...X, be 'n' random variable which are identically distributed such that Xg= 1 with a prob.…

A: Given that Probability mass function of Xk xk -1 1 2 P(XK=xk) 1/6 1/2 1/3 Where…

Q: Let X₁, Xn be a random sample from the beta distribution with a = B = 0 and = = {0: 0= 1,2}. Show…

Q: 2. Let U1, U2,... be independent random variables, each with continuous distribution that is uniform…

A: The problem can be solved using the concept of expectation and moment generating function.

Q: 1. Let X,,X,,.., be a random sample from a Geometric distribution fx(x) = p(1– p)* , x=1,2,3,... a.…

A: 1) X1,X2,X3.....,Xn are the identical independent random variables from geometric…

Q: Let {Xt; t = 0, 1, 2,...} be a birth and death process having birth rate Yn = 2(n + 1) and death…

A: i) To find the probability that the process is in state 0 in the steady state of the birth and death…

Q: 2. Let Y₁,..., Yp be independent random variables such that Y~ N(0, 1). Write Y = (Y₁...., Yp) and 0…

A: To show that an unbiased estimator of the risk of the James-Stein estimator iswe can use the…

Q: Let X1, X2, ... Xn random variables be independent random variables with a Poisson distribution…

A: Let x1 , x2 ......xn random variables be independent random variables with a Poisson distribution…

Q: Let X₁, , Xn be a random sample from a large population following a uniform distribution over [0,…

Q: 7. A discrete random variable X taking non-negative integer values has probability gen- erating…

Q: If X1 and X2 are independent random variables with distribution given by P[Xi] = −1 = P[Xi = 1] =…

A: Construct the PMF of X1X2, So, P(X1X2 = x) = P(X1=1, X2 = 1) +P(X1=-1, X2 = -1) if x =1P(X1=-1,…

Q: Suppose the random variable, X, follows an exponential distribution with mean 8 (0 ≤ @<∞). Let X₁,…

A: Random variable X follows exponential distribution with mean Random variable X follows exponential…

Q: 2. If X is a random variable taking values 0,1,2,.... and P(X) = ab*, where a and b are positive…

A: i) First part:ii) second part:

Q: R(t, t2) = 9 + 4 e-0.2 t, - t, Determine the mean, variance and covariance of the random variables Z…

Q: 6) Consider the random variable Z₁ = cos (2 * π * Yn/n) where Yn is binomial(n,p) a. What is its…

A: This question considers a random variable Zn, which is defined as the cosine of 2π times a…

Q: Let X be a uniform random variable over the interval 1 to 4 and Y is exponential with a mean of 2.…

A: Correlation Coefficient between two Random Variables: If X and Y be two random variables, then the…

Q: Suppose Y1,Y2,··· ,Yn are i.i.d. continuous uniform(0,1) distributed. (a) Prove that the kth-order…

A: Given that the iid continuous random variables Y1, Y2, . . ., Yn follow Uniform0,1 distribution.

Question

Let X₁, X₂, X₃, . . . be a sequence of independent Poisson distributed random variables with parameter 1. For n ≥ 1 let S_n = X₁ + · · · + X_n.

(a) Show that G_Xi(s) = e^s−1.
(b) Deduce from part (a) that G_Sn(s) = e^ns−n.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Trending now

This is a popular solution!

Step by step

Solved in 4 steps with 5 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

Suppose that a random vector Y' = (Y1, Y2, Y3) has a multi-normal distribution with mean vector u and variance-covariance matrix E given by 1 2 and Σ= respectively. 1 1 What is the distribution of Z = Y? 1 1
A random walk (RW) {Sn}n>o is a sum of id r.v.s X1, X2, •• , Xn,withP(X1 = a) = p, P(X1 = b) = 1 -p= q.Find the expectation E(Snl and variance var(Sn) of Sn for any n.
Show that if t" (t + t) where t and ť are both most efficient estimators with variance v, then var (t") = v (1 + p).
Let X1, X2,.., X, be independent identically distributed random variables with each X, having a probability mas function given byP(X; = 0) = 1-p P(X; = 1) = p, where Osps1. 1 EXj, then E(Y)= j = 1 Define the random variable Y = n Select one: а. 1 b. p C. 0 d.
Consider a random variable Xwith EJ X1=5 and o.’ = 3. Another random variable is given as Y =-8X+10. Find (a) E[X²] (b) E[XY] (c) E[Y²] and (d) 0,.
2. Suppose that X1, . Xn are iid Geometric random variables with frequency func- ... tion f(x; 0) = 0(1 – 0)", x = 0, 1, 2, ..., 0 E (0, 1). Find the ML estimator 0, of 0. Show that Ôn is consistent an find its asymptotic distribution.
Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?
9. If X and Y are two random variables and let g(X) be a random variable. Show that (a) E[g(X) X=x] = g(x). (b) E[g(x)Y|X=x] = g(x) E[Y|X=x]. Assume that E[g(x)] and E[Y] exist.
11. Let (y1, Y2 ... Yn) be independent random sample from the uniform distribution on [0, 1]. (a) show that Z = – In Y; has exponential distribution with parameter 1. (b) Hence or otherwise, show that -2 In Y; xản
7. Let X~ N (0,0²) and {X; : i = 1,2,..., n} be a random sample from X. (a) Formulate the log-likelihood function. (b) Find the ML estimator of o². (c) Derive the variance of the ML estimator of o2, 62. Does the variance of ô2 achieve the CR bound? (d) Derive the asymptotic distribution of √n (-o).
Suppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variables