Transcribed TextTranscribed Text

Problem 1 Let x be one observation from a N(0,0²) population. Is |X| a sufficient statistic? Problem 2 Let X1. X= be independent random variables with densities = 0 Prove that T = is a sufficient statistic for 0. Problem 3 One observation is taken on a discrete random variable x with pmf f(x|0), where OE {1,2,3}. Find the MLE of 0. x f(x|1) f(x|2) f(x)3) 1/3 1/4 1 1/3 1/4 2 1/4 1/4 3 1/6 1/4 1/2 4 1/6 1/4 Problem 4 Let X1 X. be i.i.d with Xi n Poisson(A), r > 0. Compute the ML estimator for A. Problem 5 Assume that a gas station needs at least a > 0 minutes for an oil change, where a is some constant. The actual time that an oil change requires varies from case to case. Suppose you want to model the required time for an oil change by a random variable x = Z+a, where Z is a random variable that satisfies Z 0. Suppose that Z N Exponential (1), i.e., the density of Z is given by (a) What is the density of X? 1 (b) Suppose that you have 10 independent observations of times required for an oil change: 1.2,3.1,3.6,4.5,5.1,7.6,4.4,3.5,3.8,4 Based on these data, compute the ML estimate for a. Problem 6 During the preparation for an exam Bob and Anna communicate via e-mail to clarify arising questions. Due to the time points that Anna receives e-mails from Bob she is afraid that he works more than she does. The time points (on a 24 hour scale) of the last ten e-mails are 10.55, 14.9, 11.2, 18.85, 9.75, 11.5, 16.1, 14.4,9.2, 12.95; Based on these data Anna wants to estimate the daily working time of Bob. For this pur- pose she assumes that Bob starts studying each day at a fixed time a and works until a fixed time b > a. Furthermore, she assumes that the time points when Bob sends e-mails are inde- pendent and uniformly distributed on the interval [a, bj. Can you help Anna and estimate for her the parameters a and b with (a) ML estimation; (b) The method of moments, i.e., find those parameters a, b for which the mean and the vari- ance of a Ua, b] distribution coincide with the sample average and the sample variance, respectively. What do you think about these estimation procedures? 2

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

    $50.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Probability Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats