2. It can be shown that if the random variable Trepresents the time to the rth event in a Poisson process with rate parameter 1, then T has a Gammadistribution with parameters r and 2. That is, T~r(r, 1). Assuming T~F(2,1), find P(T < 2) in each of the following ways:By integrating the Gamma pdf from 0 to 2 (recall r(r) = (r – 1)! When r is an integer). See section 4.8 if you don't recall the pdf.Using the formula F(x) = P(T < x) = 1- E=} e¬Ax CAа.b.where T r(r,1).

Answered: Image /qna-images/answer/fd093b0d-abea-4b16-8067-ee920a10f865.jpg

2. It can be shown that if the random variable Trepresents the time to the rth event in a Poisson process with rate parameter å, then T has a Gamma distribution with parameters r and A. That is, T~r(r, 1). Assuming T~[(2,1), find P(T < 2) in each of the following ways: By integrating the Gamma pdf from 0 to 2 (recall r(r) = (r – 1)! When r is an integer). See section 4.8 if you don't recall the pdf. Using the formula F(x) = P(T

2. It can be shown that if the random variable Trepresents the time to the rth event in a Poisson process with rate parameter å, then T has a Gamma distribution with parameters r and A. That is, T~r(r, 1). Assuming T~[(2,1), find P(T < 2) in each of the following ways: By integrating the Gamma pdf from 0 to 2 (recall r(r) = (r – 1)! When r is an integer). See section 4.8 if you don't recall the pdf. Using the formula F(x) = P(T

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

Similar questions

Consider independent observations (rı, y1), ... (rns yn), from the model Y, Bin(ri, p) for i = 1, ... , n, where the r, are fixed constants. Using likelihood L(p) and log-likelihood I(p) as appropriate, compute the following items. N 1. Derive the maximum likelihood estimate p. 2. Write the second derivative of log-likelihood /(p). == 3. Give an expression of the approximated asymptotic standard error of p by plugging in the estimate p. To this end, estimate the Fisher Information Matrix by : and then s. e. ap² 4. Consider data (15,11), (20,14), (15,9), (10,7), (25,17), (15,12), (10,8). Using your formulæ, compute and write numerical estimates p, s. e. (p) and give a 95% confidence interval for p using the normal approximation. e. (p) = √v¹¹ p=p Note: To answer this question you will work by hand. Do not write in the textbox but upload a single page pdf image of your workings and results. For theoretical computations (a) to (c) you are expected to show your equations and developments…
5. Let Y be the nth order statistic of a random sample of size n from Unif(0, 0). Let 0 has the prior pdf h(0) = Ba® /0B+1, 0 > a > 0, where a > 0, ß > 0. Find the Bayes estimator of 0 under the squared error loss.
Suppose that X ~ Gamma(3, b), with pdf -br f(r\b) = 2 on a > 0, where b> 0. It is required to test the null hypothesis Họ : b= bo against the alternative hypothesis H1 : b = b1, based on a random sample #1,x2,..., Tn. (a) Find the log-likelihood l(b). (b) Let LR be the likelihood ratio L(b1)/L(bo). Show that the log of the likelihood ratio is log(LR) = 3n log bo (2) - (b1 – bo) ri- i=1 (c) Using the expression for log(LR) that you obtained in part (b), show that the rejection region R= {x : log(LR) > d} for the test is of the form T bo. Hence give c' in terms of d.
49. Cars arrive at the beginning of a long road in a Poisson stream of rate from time t = 0 onwards. A car has a fixed velocity V> 0 which is a random variable. The velocities of cars are independent and identically distributed, and independent of the arrival process. Cars can overtake each other freely. Show that the number of cars on the first x miles of the road at time t has the Poisson distribution with parameter λE[V-¹ min {x, Vt}].
Given the random variable x- f(x) = ae" for x20. Estimate the moment generating function of x.
The moment generating function of the exponential distribution with rate A is m(t) = for t < A. Moreover, skewness of a random variable X is defined as X – E(X) E /var(X) Find the skewmess for the standard exponential distribution.
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,.
Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.
Suppose X and Y are independent, exponentially distributed random variables with rate parameter λ, λ > 0. Find the joint PDF of U and V , where U = X + Y, V = X/Y.
Assume the time required to fully recharge an electric car is uniformly distributed between 160 and 220 minutes. What is the expected charge time, E(x)?
3) Assume that a random variable X has moment generating function as follows: e' + e²¹ M(t)=4-2e¹ ∞ Calculate the mean and variance of X. for t
49. Cars arrive at the beginning of a long road in a Poisson stream of rate from time t = 0 onwards. A car has a fixed velocity V> 0 which is a random variable. The velocities of cars are independent and identically distributed, and independent of the arrival process. Cars can overtake each other freely. Show that the number of cars on the first x miles of the road at time t has the Poisson distribution with parameter λE[V-¹ min {x, Vt}].