Question
In: Math
Recall the lifetime (in months) of a battery is modeled by a random variable X that...
Recall the lifetime (in months) of a battery is modeled by a random variable X that has pdf fθ(x)=Kθx1(x>0)where K=ln(1/θ) for an unknown parameter θ∈(0,1) .
Assume instead that we cannot actually observe the lifetime of the batteries. Instead, we only observe if the battery is still working after τ months for some known τ to be chosen later (this is called censored data ).
Let Y1,…,Yn be our observations where Yi=1(Xi>τ) indicates that the i th battery is still working after τ months. Our goal is to estimate θ∈(0,1) (the parameter for the pdf of X ) based on this new data.
The quantity n−−√(θ~−θ) converges in distribution to N(0,σ2~) . Find the asymptotic variance σ2~ . σ2~=
Solutions
milcah answered 6 months agoRelated Solutions
For a particular machine, its useful lifetime (random variable T) is modeled by an exponential probability...
For
a particular machine, its useful lifetime (random variable T) is
modeled by an exponential probability density function. Given that
the expected lifetime of this machine is 10 years, find the
exponential probability density function f(t) for random variable
T
For a particular machine, its useful lifetime (random variable T) is modeled by an exponential probability...
For
a particular machine, its useful lifetime (random variable T) is
modeled by an exponential probability density function. Given that
the expected lifetime of this machine is 10 years, find the
exponential probability density function f(t) for random variable
T
5. The lifetime of a car battery can be modeled as a Weibull distribution with a=0.9....
5. The lifetime of a car battery can be modeled as a Weibull
distribution with a=0.9. a) If the probability that a battery works
longer than 10 years is 0.45, find the value of the parameter λ? b)
What is the time to which 75% of the battery work?
A battery manufacturer claims that the lifetime (X) of a certain type of battery is normally...
A battery manufacturer claims that the lifetime (X) of a certain
type of battery is normally distributed with a population mean of
40 hours and standard deviation 10 hours.
(a) If the claim is true, what is P ( X ≤ 36.7 )?
(b) Let X ¯ be the mean lifetime of the batteries in a random
sample of size 100. If the claim is true, what is P ( X
¯ ≤ 36.7 )?
Recall that for a random variable to be a binomial random variable, you must have an...
Recall that for a random variable to be a binomial random
variable, you must have an experiment which meets the following
three criteria:
1: There are exactly two outcomes for each trial.
2: There are a fixed number (n) of trials.
3: The trials are independent, and there is a fixed probability of
success (p) and failure (q) for each trial.
For each of the two situations described below, please indicate
if the variable X (as defined in each situation)...
Question 1. For each random variable, state whether the random variable should be modeled with a...
Question 1.
For each random variable, state whether the random variable
should be modeled with a Binomial distribution or a Poisson
distribution. Explain your reasoning. State the parameter values
that describe the distribution and give the probability mass
function.
Random Variable
1. A quality measurement for cabinet manufacturers is whether
a drawer slides open and shut easily. Historically, 2% of drawers
fail the easy slide test. A manufacturer samples 10 drawers from a
batch. Assuming the chance of failure is...
The lifetime of a particular bulb is a random variable with an average of μ =...
The lifetime of a particular bulb is a random variable with an
average of μ = 2000 hours and a standard deviation of σ = 200
hours.
(c) Calculate the probability that a bulb will last more than
2300 hours if it lasts longer than 2100 hours.
(d) Calculate the probability that a bulb will last less than 2150
hours if it has lasted more than 2050 hours.
The lifetime of a particular bulb is a random variable with an average of μ =...
The lifetime of a particular bulb is a random variable with an
average of μ = 2000 hours and a standard deviation of σ = 200
hours.
(a) What is the probability that a bulb has a lifetime between
2000 and 2400 hours?
(b) What is the probability that a bulb will last less than 1470
hours?
(c) In the case that a bulb lasts for more than 2100 hours, the
probability that it will last more than 2300 hours...
Which type of variable is the lifetime of a car battery? Categorical nominal numerical continous scale...
Which type of variable is the lifetime of a car
battery?
Categorical nominal
numerical continous scale
numerical discrete
categorical ordinal
Determine whether or not the random variable X is a binomial random variable. (a) X is...
Determine whether or not the random variable X is a binomial
random
variable.
(a)
X is the number of dots on the top face of a fair die
(b)
X is the number of hearts in a five card hand drawn (without
replacement) from a well
shuffled ordinary deck.
(c)
X is the number of defective parts in a sample of ten randomly
selected parts coming from a manufacturing process in which 0.02%
of all parts are defective.
(d)
X...
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