A certain type of electronic component has a lifetime Y
Chapter 9, Problem 84E(choose chapter or problem)
A certain type of electronic component has a lifetime (in hours) with probability density function given by
\( \mathrm{f}(\mathrm{y} \mid \theta)=\left\{\begin{array}{ll} \left(\frac{1}{\theta^{2}}\right) \mathrm{ye}^{-\mathrm{y} / \theta}, & \mathrm{y}>0 \\ 0, & \text { otherwise } \end{array}\right.\)
That is, has a gamma distribution with parameters \(\alpha=2\) and \(\theta\). Let \(\hat{\theta}\) denote the MLE of \(\theta\). Suppose that three such components, tested independently, had lifetimes of 120,130 , and 128 hours.
a Find the of \(\theta\).
b Find \(E(\widehat{\theta})\) and \(V(\hat{\theta})\).
c Suppose that \(\theta\) actually equals 130 . Give an approximate bound that you might expect for the error of estimation.
d What is the MLE for the variance of
Equation Transcription:
{
Text Transcription:
f(y \mid \theta)={(1 \theta^2) ye^-\y / \theta, y >0 0, otherwise
\alpha=2
\theta
\hat\theta
\theta
\theta
E(\widehat\theta)
V(\hat\theta)
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