A random sample of size 8 is taken from a Exp (?)

QUESTION:

A random sample of size 8 is taken from a \(\operatorname{Exp}(\lambda)\) distribution, where \(\lambda\) is unknown. The sample values are 2.74,6.41,4.96,1.65,6.38,0.19,0.52, and . This exercise shows how to use the bootstrap to estimate the bias and uncertainty \(\left(\sigma_{\hat{\lambda}}\right)\) in   \(\bar{\lambda}=1 / \bar{X}\).

a. Compute  \(\bar{\lambda}=1 / \bar{X}\) for the given sample.
b. Generate 1000 bootstrap samples of size 8 from an \(\operatorname{Exp}(\hat{\lambda})\)distribution.
c. Compute the values \(\bar{\lambda}_{i}^{*}=1 / \overline{X_{i}^{*}}\)for each of the 1000 bootstrap samples.
d. Compute the sample mean \(\bar{\lambda} *\) and the sample standard deviation \(s_{\lambda^{*}}\)of \(\widehat{\lambda}_{i}^{*}, \ldots, \widehat{\lambda}_{1000}^{*}\).
e. Estimate the bias and uncertainty \(\left(\sigma_{\lambda}\right)\)in \(\bar{\lambda}\).

Equation Transcription:

Text Transcription:

Exp(Lambda)

Lambda

Lambda hat = 1/X bar

Exp(Lambda hat)

Lambda hat sub i^* = 1/X bar sub i^*

Lambda bar *

S_lambda

Lambda hat sub i^*,...,Lambda hat sub 1000^*

(Sigma_Lambda hat)

Lambda hat