The maximum likelihood estimate of the rate parameter of exponential population based on the sample 1, 3, 1, 8, 2, 8, 3, 8, 2, 13 is:  

Key Points 

• The exponential distribution has a probability density function (pdf) given by \(f(x; λ) = λe^{(-λx)}\) for x ≥ 0 and λ > 0. The rate parameter λ is the parameter we want to estimate.
• The method of maximum likelihood estimation (MLE) is commonly used for this purpose. The likelihood function for a sample {x₁, x₂, ..., x_n} drawn from an exponential distribution is given by: \(L(λ; x_1, x_2, ..., x_n) = λn \times e^{(-λ \sum x_i)}.\)
• Taking the natural logarithm gives the log-likelihood function, which simplifies the process of finding the maximum: \(l(λ; x_1,X_2, ..., x_n) = n log (λ) - λ \sum x_i\)
• To find the maximum likelihood estimate, we differentiate this function with respect to λ, set the derivative equal to zero, and solve for λ. This gives: 
  \({d \over dλ} [l (λ; x_1, x_2, ..., x_n)] = {n \over λ} - \sum x_i = 0\)
• Rearranging this gives the maximum likelihood estimate for λ: \( λ̂ = {n \over \sum x_i}\)
• For the given sample {1, 3, 1, 8, 2, 8, 3, 8, 2, 13}, we have n = 10 (the number of observations) and ∑x_i = 1 + 3 + 1 + 8 + 2 + 8 + 3 + 8 + 2 + 13 = 49.
• Plugging these into the formula gives: \(λ̂ = {10 \over 49} \approx 0.204\)

The maximum likelihood estimate of the rate parameter λ for the given sample is approximately 0.204.