Question

Suppose that a random variable X has the exponential distribution with meanθ, which is unknown(θ >0). Find the Fisher informationI(θ)inX.

Short answer

The fisher information for the random variable X is 1/θ².

Step-by-step solution

Given the information

There is a random variable X, which follows the exponential distribution with a meanθ. Hereθ is unknown and greater than 0.

Determine the log-likelihood function

The density function of the distribution is,

F(x) = 1/θe^-x/e

Now the likelihood function is L (θ) = 1/θe^-x/e

I (θ) = ln [L (θ)]

= ln [1/θe^-x/e]

= -x/θ -ln (θ)

Derivate the log-likelihood function

by the derivation of the log-likelihood function,

∂/ ∂ θ l(θ) = ∂ / ∂ θ [-y/θ – ln(θ)]

= y/θ² – 1/θ

Now the double equation ∂²/ ∂θ² l(θ) = -2y/θ² + 1/θ²

Determine the fisher information

The fisher information is denoted as l(θ) in X

So,

I_x(θ) = -E[(∂² / ∂ θ²l(θ)]

= -E[-2y/θ³ + 1/θ²]

= 2θ/θ³ - 1/θ²

= 1/θ²

thus, the fisher information for the random variable X is 1/θ²