b)Let X₁, X₂, X3,...X₁ be a random sample of n from population X distributed with the followingprobability density function:1 -20,ef(x;0)=√2n00,if -∞0 < x < 00⁰otherwise(i) Find the parameter space of 0.(ii) Find the maximum likelihood estimator of 0.(iii) Check whether or not the estimator obtained in (ii) is unbiased.(iv) Find the Fisher information in this sample of size n about the parameter 0.

Given Xi~N(0,θ) Mean=E(X)=0, V(X)=θ Note: According to Bartleby guidelines expert solve only one…

Answered: b) Let X₁, X₂, X3X be a random sample…

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...

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2)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the maximum likelihood estimator (MLE) of parameter θ.
Find the maximum likelihood estimator for θ in the pdf f(y; θ) = 2y/(1 − θ^2), θ ≤ y ≤ 1.
Find the moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 < x < 1 0 elsewhere and use it to find μ’1,μ’2, and σ^2.
Let X1, . . . , Xn be iid with pdf f(x) = 1 x √ 2πθ2 e − (log(x)−θ1) 2 2θ2 , −∞ < x < ∞, and unknown parameters θ1 and θ2. Find the maximum likelihood estimators for θ1 and θ2, respectively
9.19 Let X and Y be two continuous random variables, with joint proba- bility density function f(x, y): - 30 -50x²-50y² +80xy for -
X1 and X2 are two discrete random variables, while the X1 random variable takes the values x1 = 1, x1 = 2 and x1 = 3, while the X2 random variable takes the values x2 = 10, x2 = 20 and x2 = 30. The combined probability mass function of the random variables X1 and X2 (pX1, X2 (x1, x2)) is given in the table below a) Find the marginal probability mass function (pX1 (X1)) of the random variable X1.b) Find the marginal probability mass function (pX2 (X2)) of the random variable X2.c) Find the expected value of the random variable X1.d) Find the expected value of the random variable X2.e) Find the variance of the random variable X1.f) Find the variance of the random variable X2.g) pX1 | X2 (x1 | x2 = 10) Find the mass function of the given conditional probability.h) pX2 | X1 (x2 | x1 = 2) Find the mass function of the given conditional probability.i) Are the random variables X1 and X2 independent? Show it. The combined probability mass function of the random variables X1 and X2 is below
If two random variables X1 and X2 have the joint density function given by f (x1, x2) = x1x2, 0 < x1 < 1, 0 < x2 < 2 0, otherwise Find the probability that (a) Both random variables will take on values less than 1 (b) The sum of the values taken on by the two random variables will be less than 1.
The density of a random variable X is f(x) = C/x^2 when x ≥ 10 and 0 otherwise. Find P(X > 20).
X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2
If X1, X2, ... , Xn constitute a random sample of size n from an exponential population, show that X is a consis-tent estimator of the parameter θ.
Let X1, ..., Xn be a random sample of size n from a gamma population given by the density f(x; α, β) = ( x α−1e − x β βαΓ(α) if x > 0, α > 0, β > 0 0 if x ≤ 0 with Γ(α) = R ∞ 0 x α−1 e −xdx the well-known gamma distribution. 1. If α > 0 is known for this random sample compute the maximum likelihood estimator βbML estimator of the unknown parameter β > 0. 2. Compute the moment generating function E(e sX) for every s < 1 β . 3. Compute the expectation E(X). 4. Show that the maximum likelihood estimator βbML is a unbiased estimator of β.(Hint: you may use the result of part 2)
If the random variable T is the time to failure of a commercial product and the values of its probability den-sity and distribution function at time t are f(t) and F(t), then its failure rate at time t is given by f(t)1 − F(t). Thus, thefailure rate at time t is the probability density of failure attime t given that failure does not occur prior to time t.(a) Show that if T has an exponential distribution, thefailure rate is constant. (b) Show that if T has a Weibull distribution (see Exer-cise 23), the failure rate is given by αβt β−1.

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