Exponential distribution

Link: Probability distribution

What’s exponential distribution? §

The exponential distribution is commonly used to model time.

PDF of exponential distribution §

$$f(x)=\lambda e^{-\lambda x}$$

where λ is the rate of how quickly it happens. A larger λ represents a steeper line, while smaller λ represents a slower line.

PDF in graph §

Maximum likelihood in exponential distribution §

The goal of Maximum likelihood is to find an optimal value for λ.

This is the likelihood of λ given the data x1, x2, …xn. Check more details about the math derivation in this video.

$$L(\lambda \mid x_1,x_2,...,x_n)={\lambda }^n[e^{-\lambda (x_1+x_2+...+x_n)}]$$

Based on above equation, we can try different λ to find the best fit by *taking the derivative of above equation = 0 and solve for λ as below. Again, check this video to learn the math behind it. After that, we will get the maximum likelihood estimate for λ:

$$\lambda =\frac{n}{(x_1+x_2+...+x_n)}$$