Exponential Likelihood

Exponential Distribution Log-Likelihood

Probability Density Function

$$f(x\mid \lambda )=\lambda e^{-\lambda x},x\ge 0$$

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Log-Likelihood

Single Observation

Given one observation ( x ):

$$\ell (\lambda \mid x)=\log \lambda -\lambda x$$

Sample of ( n ) Observations

For independent observations $x_1,\dots ,x_n$:

$$\ell (\lambda \mid x_1,\dots ,x_n)=\sum _{i=1}^n\log (\lambda e^{-\lambda x_i})=n\log \lambda -\lambda \sum _{i=1}^n{x}_i$$

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Maximum Likelihood Estimator (MLE)

Taking the derivative and setting it to zero:

$$\hat{\lambda }=\frac{n}{{\sum }_{i=1}^n{x}_i}=\frac{1}{\bar{x}}$$

See also: Exponential Distribution