Maximum likelihood estimation
The random variables X1,X2,…,Xn are continuous, independent, and distributed according to the Erlang PDF
fX(x)=[λ^3*x^2*e^(−λx)]/2, for x≥0,
where λ is an unknown parameter. Find the maximum likelihood estimate of λ, based on observed values x1,x2,…,xn . Express your answer as a function of n and s where s=x1+x2+…xn .
λ^ML=..?
3*n/s
Anyone has a clue about this?
To find the maximum likelihood estimate (MLE) of λ based on the observed values x1, x2, ..., xn, we need to maximize the likelihood function given the data.
The likelihood function L(λ) is defined as the joint probability density function (PDF) of the observed values, assuming the variables are independent and identically distributed (i.i.d). In this case, as the random variables Xi are independent and exponentially distributed with parameter λ, the joint PDF can be written as:
L(λ) = f(x1) * f(x2) * ... * f(xn)
Taking the logarithm of the likelihood function makes the maximization simpler:
log(L(λ)) = log(f(x1)) + log(f(x2)) + ... + log(f(xn))
Now, substituting the PDF of the Erlang distribution (given in the problem) for each fi(xi), we have:
log(L(λ)) = log([λ^3 * x1^2 * e^(-λx1)]/2) + log([λ^3 * x2^2 * e^(-λx2)]/2) + ... + log([λ^3 * xn^2 * e^(-λxn)]/2)
Using the properties of logarithms, we can simplify this expression:
log(L(λ)) = 3log(λ) + 2log(x1) - λx1 + 3log(λ) + 2log(x2) - λx2 + ... + 3log(λ) + 2log(xn) - λxn - nlog(2)
Combining like terms, we get:
log(L(λ)) = 3nlog(λ) + 2log(x1x2...xn) - λ(x1 + x2 + ... + xn) - nlog(2)
To find the MLE, we need to maximize this log-likelihood function with respect to λ. Taking the derivative of log(L(λ)) with respect to λ and setting it equal to zero, we can solve for the maximum likelihood estimate:
d/dλ (log(L(λ))) = 0
Differentiating the log-likelihood function, we have:
3n/λ - (x1 + x2 + ... + xn) = 0
Simplifying the equation:
3n/λ = (x1 + x2 + ... + xn)
Solving for λ, we get:
λ^ML = 3n / (x1 + x2 + ... + xn)
Expressed in terms of n and s (where s = x1 + x2 + ... + xn), the maximum likelihood estimate of λ is:
λ^ML = 3n / s