Graph Theory/ Maximum Likelihood/ Probability

Question

Suppose we draw n independent random reals x in the range 0≤x≥∞ from the (properly normalized) exponential probability density P(x)=µe^-µx.

a) Write down the likelihood (i.e. the probability density) that we draw a particular set of n values x_i (where i=1...n) for a given value of the exponential parameter µ.

b) Hence, find a formula for the best (meaning the maximum likelihood) estimate of µ given a set of observed values x_i

Answer 1

a) For this problem, x-bar is the sample mean

P(x₁,x₂, ... ,x_µn) = µe^-µx1 * µe^-µx2 * ... * µe^-µxn = µⁿe^-µ∑xi = µⁿe^{-nµ x-bar}

b) It is easier to maximize the log likelihood = n ln µ - n µ x-bar

d ln(P)/dµ = n/µ - n x-bar

n/µ - n x-bar = 0

n/µ = n x-bar

n µ x-bar = n

µ = 1/x-bar