To find the MLE of the parameter "a" in the given sample from Exp(X), we use the likelihood function and differentiate it with respect to "a". By solving the resulting equation, we find that the MLE of "a" is approximately 0.1837.
To determine the maximum likelihood estimate (MLE) of the parameter "a" in the given sample from Exp(X), you can use the following steps:
1. The exponential distribution can be expressed as: f(x|a) = a * exp(-ax), where "a" is the parameter of interest.
2. The likelihood function, L(a), can be calculated by taking the product of the density function values for each observation in the sample. In this case, L(a) = a^n * exp(-aΣx), where "n" is the sample size and Σx represents the sum of all observations in the sample.
3. To find the MLE, we need to find the value of "a" that maximizes the likelihood function.
Taking the natural logarithm of L(a), we get: ln(L(a)) = nln(a) - aΣx.
4. To find the MLE, we differentiate ln(L(a)) with respect to "a" and set it equal to 0: d/d(a)(ln(L(a))) = n/a - Σx = 0.
5. Solving the equation for "a", we get: a = n/Σx.
6. Substituting the values from the given sample into the formula, we have: n = 10 (since there are 10 observations) and Σx = 1,76 + 8,62 + 6,43 + 4,73 + 10,01 + 0,18 + 2,62 + 3,40 + 4,88 + 11,81 = 54,44.
7. Plugging in the values, we find that a = 10/54.44 ≈ 0.1837.
Therefore, the MLE of "a" in the given sample is approximately 0.1837.