Answered: Assume that survival times are…

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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Assume the data are generated from the GN-SLR setting, Y = B1Xin + ti i = 1, (a) Show that the likelihood function is where €; ~ N (0,0²). We only have the observed versions of (Y₁, X₁1), which are (x₁1,₁),..., (Xn1, Yn). We want to estimate B₁ and o² with these data using the Maximum Likelihood Method. n i=1 1 L(Bo, B₁,0²) = (2ño²)−¹/² exp{-22 (Yi - B₁x₁)²} (b) Compute the negative log-likelihood function. (c) Show that the MLE estimator for ₁ is ., n 9.9 1(Bo, B₁,0²) = -log L(Bo, B₁, 0²) ÂMLE _ Σi=1 X₁Yi = An 2 (d) Show that the MLE estimator for o2 is i=1 - (62) MLE _ Σi=1(Yi – ÎMLEƑ¡1)² n
A continuous random variable X can take only values in the interval [0, 4]. In this interval, the probability that X takes a value greater than x is equal to ax^2 + b. (i) Determine the values of a and b. (ii) Find the cumulative distribution function F(x).
A random sample of n volumes in the form of X1, X2 . , Xn is .. taken from a population with parameters 0, and 02 with a continuous uniform distribution. Let the 0, parameter be known. Based on this random sample, obtain the Maximum Likelihood Estimator ^@Mle of parameter 02.
The time to failure (in hours) of a piece of equipment is uniformly distributed over (0, 1000) hours. (a) Determine the MTTF. (b) Determine the MTTF if preventive maintenance will restore the system to as good as new condition and is performed every 100 operating hours. (c) Compare the reliability with and without preventive maintenance at 225 operating hours. Assume the 100 hour maintenance interval and a maintenance-induced failure probability of 0.01 each time preventive maintenance is performed.
An accident-prone motorist has two tragic accidents with two weeks between rooms. The 'waiting time' (measured in days) between two accidents can be approximately approximated as exponentially distributed with unknown expected value µ. Determine a 90% condensation interval for µ. Please explin so I understand to ba able to solve similar kind of questions..
The time until a chemical reaction is complete (in milliseconds) is approximated by cumulative distribution function F(2) = { ,"ani 0lz The probability that a reaction completes within 400 milliseconds is
How do find the E[X^2].
For a GLM with canonical link function, explain how the likelihood equations imply that the residual vector e = (y – @) is orthogonal with C(X).
Exercise 10. For an exponential random variable X (Definition 10), show that E[X] = }. More generally, for a function o : R R, E(¢(X)] = | (u) dF(u) (5.4)
Suppose that X₁ = 1; X₂ = 1; X3 = 0; X₁ = 1; X5 = 1, X6 = 1; X₂ = 0; X = 1; X9 = 0, X10 = 0, represents a random sample. Each of these X's comes from the same population and has a density of fx₁ = 0¹-* (1 - 0)*; x = 0,1 First determine the form of the maximum likelihood estimator for 0, then use the MLE formula and the data provided to find an estimate for 0. Round your answer to 3 decimal places. (Answer is 0.4)
The probability density function of the random variable X is as in the picture with λ> 0. Investigate whether the most likelihood estimator (λ^) of λ is neutral. hint : ?(?) = √??/2 and ?(?²)=2?
Suppose X1, ..., Xn have been randomly sampled from a normal distribution with mean 0 and unknown variance sigma^2, and let U = c * i=1 -> n summation (X_ i)^2 , where c is a constant. Find the value of c that minimises the Mean Squared Error (MSE)

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Assume that survival times are exponentially distributed S(t) = e^(-ut). Compute the following and estimate u y maximum likelihood