**Problem Statement:**Suppose $ X_1, \ldots, X_n $ are independent and identically distributed (iid) with probability density function (pdf) \[f(x; \theta) = \frac{1}{\theta} e^{-x/\theta}, \quad 0 < x < \infty,\]zero elsewhere.**Tasks:**(a) Find the Maximum Likelihood Estimator (MLE) of $\theta$.(b) Find the MLE of $ P_\theta(X > 1) $.(c) Prove that the MLE you find in (b) is a consistent estimator of the probability $ P_\theta(X > 1) $.

Answered: Image /qna-images/answer/c649ac7f-7a63-4d81-aa07-1ac75882b27e.jpg

Answered: (a) Find the MLE of 0. (b) Find the MLE…

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

(a) Determine the value of k for a valid pdf.(b) Complete the cdf of X .(c) What is the expected total deduction on a randomly chosen form?(d) Let Y = ln(X /5), where ln(·) is natural logarithm. Determine the pdf as wellas the cdf of Y , and further identify which common distribution that Y follows?
What is the likelihood ratio p(x|C₁) p(x|C₂) in the case of Gaussian densities?
Let Y be the present value random variable of a 10-year term annuity-due of 1 issued at an individual (35). Calculate its actuarial present value if (a) Mortality follows De Moivre's Law with w = 100. (b) 8 = 0.05.
(b) The probability density of a function is given by p(x) from x = 1 to x = 4. That is 62 x3 dx = 1 62 P(x) dx = - 00 Determine the value of the standard deviation.
(i) and (ii) please, thank you
Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.
5. Let X be a positive random variable; i.e., P(X - log(EX) (b) E[log(1/X)] > log[1/EX] (c) E (X³) > (EX)³
The time intervals between successive barges passing a certain point on a busy waterway have an exponential distribution with mean 10 minutes. (a) Find the probability that the time interval between two successive barges is less than 8 minutes. (b) Find a time interval t such that we can be 95% sure that the time interval between two successive barges will be greater than t.
A individual, with a utility function lnW and an initial wealth level W0, is faced with a fair gamble of winning or losing $h (where W0 > h > 0) with 50-50 chance. (a) Is this individual risk averse? Explain.(b) Suppose that the individual is willing to pay up to an amount of f in order to avoid such a gamble. Give the equation that determines f, and solve the equation for f (i.e., express f in terms of W0 and h). (c) Show that f increases as h increases.
The probability density function of the time it takes a hematology cell counter to complete a test on a blood sample is f(x) = 0.04 for 58 < x< 83 seconds. Round your answers to 2 decimal places. (a) What proportion of tests require more than 70 seconds to complete? i (b) What proportion of tests require less than one minute to complete? i (c) Determine the mean and variance of the time to complete a test on a sample. Mean = i seconds Variance = i seconds?

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

(a) Find the MLE of 0. (b) Find the MLE of Po(X > 1). (c) Prove that the MLE you find in (b) is a consistent estimator of the probability Po(X > 1).