**Problem 3.** Recall that if \( X \) follows a Poisson distribution with parameter \( \lambda \), the probability mass function of \( X \) is given by:\[p_X(x) = \frac{e^{-\lambda} \lambda^x}{x!}, \quad x = \{0, 1, 2, \ldots\}\](a) If \( p_X(2) = 2p_X(0) \), calculate \( p_X(3) \).(b) Use this Poisson random variable \( X \) to approximate \( \mathbb{P}(Y = 4) \), where \( Y \) is a binomial random variable with \( n = 2000 \) and \( p = \lambda/n \).

A binomial distribution can be approximated by a Poisson distribution under the following…

Answered: Problem 3. Recall that if X follows a…

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

1. Suppose X = (X₁,..., Xn) is a random sample from the Gamma distribution with shape a and rate 0, i.e., fx(x) = Ja r(a) X² - -x0 Each X, has expectation E(X) generating function Mx (t) = (1 – t/0)¯¤. 2 a/0, variance Var(X) x>0, a > 0, 0 >0. S (b) Verify that each score function has zero expectation. (c) Assuming a to be known: (a) Assuming both a and to be unknown, write down the log likelihood function l(0, a; X) and the corresponding score functions and де де 20 θα = (i) Calculate the Cramer Rao Lower Bound (CRLB) for the variance of an unbiased estimator of 0. (ii) Is there any unbiased estimator of whose variance attain the CRLB? 111 Show that a 1 = - -'Σ(;) i=1 a/0², and moment is an unbiased estimator for 0. What is the MVU estimator for 0? (iv) Identify a change of parameter n = n(0) for which there exists an unbiased estimator with variance attained the CRLB. (v) For the parameter in the previous part, identify the MVU estimator whose variance attains the CRLB. Compute this…
(5) A discrete random variable X has the probability mass function f(x) = c(x + 1)², x=-1,0, 1. You must show work when answering the following questions. (a) Determine the value of the constant c. (Please double check your answer. We will not give partial credit if you don't get this part right.) (b) Find the mean and the variance of X. (c) Give a formula or a sketch of F(x), the cumulative distribution function of X.
4. Question 4: Y is a discrete random variable with PMF S1/3, y=1 p(y) = 2/3, y = 2 (a) Find the MGF for Y. (b) Develop the first two raw moments from MGF and find the variance V(Y). Answers:
2. Let X₁, X2, ..., Xn represent a random sample from a Poisson distribution with probability mass function given below. p(x) = e-11e0 el x! summary sheet on Moodle and state E(X). X 2 (a) Look up the Special Distributions (b) Find the moment estimator of X. (c) Show that the maximum likelihood estimator of X is the same as the moment estimator. x = 0, 1, 2, ... (d) Estimate A from the following observations using the moment/maximum likelihood. estimator. Round your answer to 4 decimal places. Frequency 18 0 1 2 3 4 5 6 7 37 42 30 13 7 2 1
E2
1. The time to failure of an AC compressor is a random variable T, with the following pdf. 200 f(t) = (10 + t)3 (a) Find R(t). (b) Determine the reliability for the first year of operation. (c) Find the probability of failure occurring in the interval of time [15,20]. (d) Find the design life if a reliability of 0.99 is desired.
1. Let X be a Poisson random variable on the non-negative integers with rate λ = 4. Let W = 2X + 10. (a) What is the range of W? (b) Find a formula for Pw(k).
B2. (a) Consider X₁,..., Xn to be a random sample from the geometric distribution, with probability mass function: P(X= x) = p(1-p), with x = : 0, 1, 2, 3,..., and p € (0, 1]. (i) Using the MGF (M(t) == (ii) Find the Maximum Likelihood Estimator (MLE) for p. -(1-p)et (b) Suppose X₁,..., X₁ is a random sample from a Beta(01, 1) population, and Y₁,..., Ym is an independent random sample from a Beta(02, 1) population. We want to find the approximate Likelihood Ratio Test for Ho: 01 02 00, versus H₁: 01 02. To this aim: = = (i) Under the alternative hypothesis H₁0₁02, show that the MLE for ₁ and 02 are: 0₁ 02 derive E[X] and Var[X]. n Σlog(x)' Recall, that the PDF of Beta(a, b) is fy (y) 00 = - m Σ log(yi) = ['(a) = (a − 1)! for all positive integers a and I (1) = 1) [(a+b) F(a) (b)-1(1-y)b-1 and (ii) Under the null hypothesis Ho: 0₁ 02 = 0o, show that the MLE for 00 is: 01 = n + m E log(xi) + log(yi)
7. a) Suppose that X is a uniform continuous random variable where 0
#2: Suppose a random variable X has expected value E[X] = 3 and variance Var(X) = 4. Compute the following quantities. (a) E[3X + 2] (b) E[X²] (c) E[(3X+2)²] (d) Var[2X + 2]
Do #2
2. The cumulative distribution function of random variable X is 1 ,x > 1 Kx Fx(x) ,0 < x < 1 x2 + 1 ,x < 0 (a) Find K. (b) Find the p.d.f fx(x). (c) The random variable Y is the output of the clipper: 1 ,X 2 1 1 Y = { 0.5 2 1 ,0 < X <- -1 4 What is E{Y}? (d) Find the characteristic function of Y. (Øy(w) =?) Hint: Y can be considered as a discrete r.v.

SEE MORE QUESTIONS

Recommended textbooks for you

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

SEE MORE TEXTBOOKS