Fisher Information of Exponential Distribution

• 𝐼(𝜆) = 𝑛/𝜆²

Derivation

the generic Fisher Information formula is as follows:

• 𝐼(𝜃) = - 𝐄[log-likelihood-function”(𝜃)]

we need the log-likelihood function 𝐿𝐿(𝜃) of an Exponential(𝜆) Distribution which is as follows

• 𝐿𝐿(𝜆) = 𝑛·𝑙𝑛(𝜆) - 𝜆·𝛴_{1≤𝑖≤𝑛} 𝑋_𝑖# click here for step-by-step computation

now we need to compute the second derivative of 𝐿𝐿(𝜆)

• 𝐿𝐿’(𝜆) = 𝑛/𝜆 - 𝛴_{1≤𝑖≤𝑛} 𝑋_𝑖
• 𝐿𝐿”(𝜆) = -𝑛/𝜆²

plug into Fisher Information formula

• 𝐼(𝜃) = - 𝐄[log-likelihood-function”(𝜃)]
• 𝐼(𝜆) = - 𝐄[log-likelihood-function”(𝜆)] # the parameters (𝜃) of exponential distribution is a single 𝜆
• 𝐼(𝜆) = - 𝐄[-𝑛/𝜆²]
• 𝐼(𝜆) = - (-𝑛/𝜆²) # expected value of a constant is itself
• 𝐼(𝜆) = 𝑛/𝜆²