Question

The observation sample of the envelope of a received signal is given by the following exponential distribution:

fYk (yk)={θ/1 exp (-yk/θ), yk≥0, k = 1, 2,..., K
{0, elsewhere
where θ is an unknown (positive) parameter and the observations are statistically independent.
Obtain the ML estimate of θ.

Answer 1

The maximum likelihood estimate (MLE) of the parameter θ for an exponential distribution, based on independent observations, is calculated by taking the product of the individual densities, deriving the likelihood function, and differentiating to find the MLE as the sample mean of the observations.

Step-by-step explanation:

The problem involves finding the maximum likelihood estimate (MLE) of the parameter θ for an exponential distribution. Since the observations are independent, the joint probability density function of the observations Y1, Y2, ..., Yk is the product of the individual densities. The likelihood function L(θ) for a sample Y1, Y2, ..., Yk is given by:

L(θ) = Π fYk(yk) = Π (θ-1exp(-yk/θ))
= θ-kexp(-Σyk/θ)

To find the MLE of θ, we take the natural logarithm of the likelihood function to get the log-likelihood function, which is easier to differentiate:

ln(L(θ)) = -k ln(θ) - Σyk/θ

Now, differentiate the log-likelihood with respect to θ and set it equal to zero to find the critical point(s). This gives us:

d/dθ[ln(L(θ))] = -k/θ + Σyk/θ2

Setting this derivative equal to zero and solving for θ provides us with the MLE for θ:

0 = -k/θ + Σyk/θ2 ⇒ θ = Σyk/k

Therefore, the maximum likelihood estimate of θ is the sample mean of the observations.