Question

Suppose that \({X_1},...,{X_n}\) form a random sample from the exponential distribution with unknown mean μ. Describe a method for constructing a confidence interval for μ with a specified confidence coefficient \(\gamma \left( {0 < \gamma < 1} \right)\)

Short answer

\(P\left( {\frac{1}{{{c_2}}}\sum\limits_{i = 1}^n {{X_i}} \le \mu \le \frac{1}{{{c_1}}}\sum\limits_{i = 1}^n {{X_i}} } \right)\)

Step-by-step solution

Given information

\({X_1},...,{X_n}\) form a random sample from the exponential distribution with unknown mean μ. We need to find a method for constructing a confidence interval for μ with a specified confidence coefficient \(\gamma \left( {0 < \gamma < 1} \right)\)

Step-2: Method for constructing a confidence interval for μ with a specified confidence coefficient \(\gamma \left( {0 < \gamma  < 1} \right)\)

The sum of n random variables with gamma distribution has a gamma distribution. Since the exponential distribution parameters \(\alpha = 1,\beta = \frac{1}{\mu }\) then the sum \(\sum\limits_{i = 1}^n {{X_i}} \) will have gamma distribution with parameters \({\alpha _1} = n,{\beta _1} = 1\) ,but from the definition it follows that the following random variable with chi-square distribution , it follows that the random variable with parameters m=2n.

However, an interval with limits taken from the following

\(P\left( {{c_1} < U < {c_2}} \right) = P\left( {\frac{1}{{{c_2}}}\sum\limits_{i = 1}^n {{X_i}} \le \mu \le \frac{1}{{{c_1}}}\sum\limits_{i = 1}^n {{X_i}} } \right)\) where U has chi-square distribution with 2n degrees of freedom.