Let X1, X2, Xn be a random sample from the uniform distribution on the interval [0,0], where > 0 is an unknown parameter. The probability density function is: f(x|0) = = { 1 0 ≤ x ≤ 0 otherwise (a) (4 points) Find the maximum likelihood estimator (MLE) for 0 based on the sample. (b) (6 points) Derive a 100(1 - α) % confidence interval for 0 based on the distribution of the maximum observation X(n) = max{X1, X2,...,- ‚ Xn}.
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help me with ab please. please handwrite if possible. please don't use AI tools to answer
![Let X1, X2, Xn be a random sample from the uniform distribution on the interval [0,0],
where > 0 is an unknown parameter. The probability density function is:
f(x|0) =
=
{
1
0 ≤ x ≤ 0
otherwise
(a) (4 points) Find the maximum likelihood estimator (MLE) for 0 based on the sample.
(b) (6 points) Derive a 100(1 - α) % confidence interval for 0 based on the distribution of the
maximum observation X(n) = max{X1, X2,...,-
‚ Xn}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6b7a6d6-eb34-4148-b819-2ad3c5088444%2Fbdf1262f-129b-4e5f-b256-a19b1dfc92d9%2F440og84_processed.jpeg&w=3840&q=75)
Step by step
Solved in 2 steps with 3 images
- Let the random variable X be the time in seconds between incoming telephone calls at a busy switchboard. Suppose that a reasonable probability model for X is given by the pdf: fx(x) = { ie for 0Random variables Y₁,... Yn follow the probability density distribution: f(y|0) = { v with known expected values E[Y] = 20 and Var[Y] = 202. a.) Find the maximum likelihood estimate of 0. b.) Find E[] and V(A) y >0 otherwise c.) Is this a biased estimator?Let (x1,x2,...,xn) be i.i.d. samples from a random variable X with the following probability density function: (-1²-μ) 1 = exp 20 fx(x) = x≤R, where ER and o> 0 are unknown. Find the maximum likelihood estimate of u and o.The differentiation approach to derive the maximum likelihood estimator (mle) is not appropriate in all the cases. Let X₁, X2,,X₁ be a random sample of size n from the population of X. Consider the probability function of X fe-(2-0), if 0The continuous random variable Y has the following pdf: 1- y?, 0Let x be a continuous random variable with PDF 2x 0 < x < 1 fx(æ) = { otherwise Find the expected value of x.A factory is supplied with grain at the beginning of each week. The weekly demand, X thousand tonnes, for grain from this factory is a continuous random variable having the probability density function given by f(x) = {2(1-x) lo 0≤x≤ 1, otherwise. Find (i) the mean value of X, (ii) the variance of X, (ii) the quantity of grain in tonnes that the factory should have in stock at the beginning of a week, in order to be 98% certain that the demand in that week will be met.Suppose the random variable, X, follows a geometric distribution with parameter 0 (0 < 0 < 1). Let X₁, X2,,X₁ be a random sample of size n from the population of X. (a) Write down the likelihood function of the parameter. (b) Show that the log likelihood function of depends on the sample only through Σj-1Xj.Let X1, X2,...,X, be a random sample from a distribution with density function e if x > 0 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?Suppose an electric-vehicle manufacturing company estimates that a driver who commutes 50 miles per day in a particular vehicle will require a nightly charge time of around 1 hour and 30 minutes (90 minutes) to recharge the vehicle's battery. Assume that the actual recharging time required is uniformly distributed between 70 and 110 minutes. (a) Give a mathematical expression for the probability density function of battery recharging time for this scenario. f(x) = , 70 ≤ x ≤ 110 , elsewhereLet the random variable X be defined on the support set (1,2) with pdf fX(x) = (4/15)x3, Find the variance of X.Let x₁ = 6, x2 = 4, x3 = 5, x4 = 15 be a random sample from a population with probability density function a 10a (x + 10)α+1 Compute â, the maximum likelihood estimate of a. f(x) = You do not have to verify that your estimate is a maximum. 2 x > 0.Recommended textbooks for youGlencoe Algebra 1, Student Edition, 9780079039897…AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillGlencoe Algebra 1, Student Edition, 9780079039897…AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill