We have a sample X1, X2, X3 of independent variables from a distribution with densityfa(x) = x for x E (0, 2) and 0 otherwise, where a > 0 is an unknown parameter.%3DFind the ML estimate of parameter a if our sample consists of observations:1.1.9.Answer:

Answered: We have a sample X1, X2, X3 of…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)
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The lifetime, X, of a particular integrated circuit has an exponential distribution with rate of ?=0.5 per year. Thus, the density of X is:f(x,?) = ? e−?x for 0 ≤ x ≤ ∞, ? = 0.5 . ? is what R calls rate. Hint: This is a problem involving the exponential distribution. Knowing the parameter ? for the distribution allows you to easily answer parts a ,b ,c and use the built-in R functions for the exponential distribution (dexp(), pexp(), qexp()) for other parts . Or (not recommended) you should be able to use the R integrate command with f(x) defined as above or with dexp() for all parts.a) What is the expected value of X? b) What is the variance of X? c) What is the standard deviation of X? d) What is the probability that X is greater than its expected value? e) What is the probability that X is > 5? f) What is the probability that X is > 10? g) What is the probability that X > 10 given that X > 5? h) What is the median of X?
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We have a sample X1, X2, X3 of independent variables from a distribution with density fa(x) = x"- for x E (0, 2) and 0 otherwise, where a > O is an unknown parameter. Find the ML estimate of parameter a if our sample consists of observations: 1.1.9. 20 Answer: