The lifespan of a system component follows a Gamma distribution with β (unknown) and α=1. (hint:...

The lifespan of a system component follows a Gamma distribution with β (unknown) and α=1. (hint: start with confidence interval for μ)

f(x)=(1/β^α Γ(α)) X^α-1 e^-X/β

a. Derive 98% large sample confidence interval for β.

b. Find maximum likelihood estimator of β.

a) 98% large sample confidence interval fo β is

b) MLE for β is

Expert Solution

orchestra answered 1 year ago

The lifespan of a system component follows a Weibull distribution with α (unknown) and β=1. (hint:...

The lifespan of a system component follows a Weibull distribution with α (unknown) and β=1. (hint: start with a confidence interval for μ) f(x)=αβXβ-1 exp(-αXβ) a. Derive 92% large sample confidence interval for α. b. Find the maximum likelihood estimator of α.

The lifespan of a system component follows a Weibull distribution with α (unknown) and β=1. (hint:...

The lifespan of a system component follows a Weibull distribution with α (unknown) and β=1. (hint: start with confidence interval for μ) f(x)=αβXβ-1 exp(-αXβ) a. Derive 98% large sample confidence interval for α. b. Find maximum likelihood estimator of α.

Suppose X follows a Gamma distribution with parameters α, β, and the following density function f(x)=...

Suppose X follows a Gamma distribution with parameters α, β, and the following density function f(x)= [x^(α−1)e^(−x/ β)]/ Γ(α)β^α . Find α and β so that E(X)= Var(X)=1. Also find the median for the random variable, X.

1. Homework 4 Due: 2/28 Let X1,...,Xn i.i.d. Gamma(α,β) with α > 0, β > 0...

1. Homework 4 Due: 2/28 Let X1,...,Xn i.i.d. Gamma(α,β) with α > 0, β > 0 (a) Assume both α and β are unknown, find their momthod of moment estimators: αˆMOM and βˆMOM. (b) Assume α is known and β is unknown, find the maximum likelihood estimation for β. 2.

If a random variable Xhas the gamma distributionwith α= 2and β = 1, (a) What is...

If a random variable Xhas the gamma distributionwith α= 2and β = 1, (a) What is the probability density functionf(x)? (b) find P(1.8 < X< 2.4). (c) What isE(X) and Var(X)? (d) Put α= 1and β = 2.What isthe probability density functionf(x). (e) What is the name of the distributionin (d)?

The lifetime of an electronic component has a Weibull distribution with parameters α=0.60 and β=2. Compute...

The lifetime of an electronic component has a Weibull distribution with parameters α=0.60 and β=2. Compute the probability the component fails before the expiration of a 4 year warranty. The maximum flood levels, in millions of cubic feet per second, for a particular U.S. river have a Weibull distribution with α=5/3 and β=3/2. Find the probability that the maximum flood level for next year will exceed 0.7 million cubic feet per second.

Let α, β be cuts as defined by the following: 1) α ≠ ∅ and α...

Let α, β be cuts as defined by the following: 1) α ≠ ∅ and α ≠ Q 2) if r ∈ α and s ∈ Q satisfies s < r, then s ∈ α. 3) if r ∈ α, then there exists s ∈ Q with s > r and s ∈ α. Let α + β = {r + s | r ∈ α and s ∈ β}. Show that the set of all cuts R with the...

The lifespan of an electrical component has an exponential distribution with parameter lambda = 0.013. Suppose...

The lifespan of an electrical component has an exponential distribution with parameter lambda = 0.013. Suppose we have an iid sample of size 100 of these components Some hints: P(X < c) for an exponential(lambda) can be found via pexp(c,lambda) E[X] = 1/lambda and Var[X] = 1/lambda^2 Round all answers to 4 decimals Using the exact probability distribution, what is the probability that a single component will be within 15.38 units of the population mean? Using Chebyshev's inequality, what is...

Consider α-diversity, β-diversity, and γ-diversity. (That is, alpha-diversity, beta-diversity, and gamma-diversity.) Over time, is habitat fragmentation...

Consider α-diversity, β-diversity, and γ-diversity. (That is, alpha-diversity, beta-diversity, and gamma-diversity.) Over time, is habitat fragmentation likely to increase or decrease each of these types of diversity within a landscape? Briefly explain your reasoning.

Let X1,...,Xn be a random sample from a gamma distribution with shape parameter α and rate...

Let X1,...,Xn be a random sample from a gamma distribution with shape parameter α and rate β (note that this may be a different gamma specification than you are used to). Then f(x | α, β) = (βα/Γ(α))*x^(α−1) * e^(−βx). where x, α, β > 0 (a) Derive the equations that yield the maximum likelihood estimators of α and β. Can they be solved explicitly? Hint: don’t forget your maximum checks, and it may help to do some internet searching...

Question

The lifespan of a system component follows a Gamma distribution with β (unknown) and α=1. (hint:...

Solutions

Expert Solution

Related Solutions

The lifespan of a system component follows a Weibull distribution with α (unknown) and β=1. (hint:...

The lifespan of a system component follows a Weibull distribution with α (unknown) and β=1. (hint:...

Suppose X follows a Gamma distribution with parameters α, β, and the following density function f(x)=...

1. Homework 4 Due: 2/28 Let X1,...,Xn i.i.d. Gamma(α,β) with α > 0, β > 0...

If a random variable Xhas the gamma distributionwith α= 2and β = 1, (a) What is...

The lifetime of an electronic component has a Weibull distribution with parameters α=0.60 and β=2. Compute...

Let α, β be cuts as defined by the following: 1) α ≠ ∅ and α...

The lifespan of an electrical component has an exponential distribution with parameter lambda = 0.013. Suppose...

Consider α-diversity, β-diversity, and γ-diversity. (That is, alpha-diversity, beta-diversity, and gamma-diversity.) Over time, is habitat fragmentation...

Let X1,...,Xn be a random sample from a gamma distribution with shape parameter α and rate...