Final answer:
To find the MLE of the median of a distribution with unknown parameter, first find the MLE of the parameter itself using the maximum likelihood estimation method, and then use it to find the MLE of the median. In this case, the random variables follow independent exponential distributions with an unknown parameter. To find the MLE of the parameter, maximize the likelihood function by taking the logarithm and solving for the MLE. Finally, use the MLE of the parameter to calculate the MLE of the median.
Step-by-step explanation:
To determine the maximum likelihood estimate (MLE) of the median of a distribution with unknown parameter, we can first find the MLE of the parameter itself and then use it to find the MLE of the median.
In this case, the random variables Xi are independent and follow an exponential distribution with parameter β (unknown). The probability density function of an exponential distribution is f(x) = βe^(-βx).
To find the MLE of β, we need to maximize the likelihood function L(β) = ∏(βe^(-βxi)), where xi are the observed values.
By taking the logarithm of the likelihood function and then maximizing it, we can solve for the MLE of β.
Once we have the MLE of β, we can use it to find the MLE of the median using the relationship between the median and the parameter of an exponential distribution: median = ln(2)/β.