Question

You have a sample consisting of the values 27, 79, 87, 45, 79, 85, and 34, which is known to come from a population with the density function:

f(x) = { θ * e^(-θx), for x > 0, and 0 otherwise.

What is the maximum likelihood estimate (MLE) of the parameter θ for this population? Please provide your answer rounded to one decimal place.

Answer 1

The MLE of the parameter θ is determined by taking the product of the density functions for each observed data point, converting this to the log-likelihood, differentiating, and solving for θ.

Step-by-step explanation:

The question asks for the maximum likelihood estimate (MLE) of the parameter θ for a population with the given density function θ * e^(-θx), for x > 0. To find the MLE of θ, we can use the likelihood function, which is the product of the density functions for each observed data point. The likelihood L(θ) for the sample is given by:

L(θ) = θ * e^{-θ(27)} * θ * e^{-θ(79)} * θ * e^{-θ(87)} * θ * e^{-θ(45)} * θ * e^{-θ(79)} * θ * e^{-θ(85)} * θ * e^{-θ(34)}

This is equivalent to:

L(θ) = θ^{7} * e^{-θ(27+79+87+45+79+85+34)}

To find the value of θ that maximizes this likelihood, we take the logarithm of L(θ) (log-likelihood), differentiate it with respect to θ, set the derivative equal to zero, and solve for θ. This procedure gives us the MLE of θ.

The final MLE value will be rounded to one decimal place as requested.