Refer to Example 8.4 and suppose that Y is a single...

Refer to Example 8.4 and suppose that $Y$ is a single observation from an exponential distribution with mean $\theta$ a. Use the method of moment-generating functions to show that $2 Y / \theta$ is a pivotal quantity and has a $\chi^{2}$ distribution with 2 df. b. Use the pivotal quantity $2 Y / \theta$ to derive a $90 \%$ confidence interval for $\theta$ c. Compare the interval you obtained in part. (b) with the interval obtained in Example 8.4

Refer to Example 8.4 and suppose that $Y$ is a single observation from an exponential distribution with mean $\theta$ a. Use the method of moment-generating functions to show that $2 Y / \theta$ is a pivotal quantity and has a $\chi^{2}$ distribution with 2 df.
b. Use the pivotal quantity $2 Y / \theta$ to derive a $90 \%$ confidence interval for $\theta$
c. Compare the interval you obtained in part. (b) with the interval obtained in Example 8.4

Key Concepts

Confidence Interval

A confidence interval is a range of values derived from sample data that is likely to contain the true value of an unknown parameter with a specified level of confidence. Confidence intervals are constructed using pivotal quantities or other statistic techniques, providing a measure of uncertainty around point estimates, which is fundamental in statistical inference and decision making.

Pivotal Quantity

A pivotal quantity is a function of the observed data and the unknown parameters that has a probability distribution independent of the parameters. This property makes pivotal quantities very useful in constructing confidence intervals and performing hypothesis tests, as they allow inferences about the parameters without knowledge of their true values.

Chi-Square Distribution

The chi-square distribution arises frequently in statistics, particularly in hypothesis testing and confidence interval construction. It is the distribution of a sum of the squares of independent standard normal random variables. In the context of pivotal quantities, demonstrating that a transformed statistic follows a chi-square distribution allows for the derivation of confidence intervals and other inferential procedures.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between independent events that occur at a constant rate. It is characterized by its mean or rate parameter, and its probability density function decreases exponentially. This distribution is fundamental in reliability analysis and various fields where waiting times are analyzed.

Moment-Generating Function

A moment-generating function (MGF) is a tool used to succinctly capture all the moments of a random variable. It is defined as the expected value of the exponential function of the random variable scaled by a parameter. The MGF is particularly useful in proving distributional properties, such as showing that a transformation of a random variable has a specific distribution, and in facilitating proofs regarding convergence in distribution.

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