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.

^λML=

^λML= (3*n)/s

To find the maximum likelihood estimate of λ, we need to maximize the likelihood function. The likelihood function is the product of the individual probabilities of observing the given values.

The probability density function (PDF) for the Erlang distribution is given by:

fX(x) = (λ^3 * x^2 * e^(-λx))/2

The likelihood function L(λ) can be written as:

L(λ) = fX(x1) * fX(x2) * ... * fX(xn)

Taking the natural logarithm of the likelihood function (lnL(λ)) will simplify the calculations and make it easier to find the maximum:

lnL(λ) = ln(fX(x1)) + ln(fX(x2)) + ... + ln(fX(xn))

For each individual observation, we substitute the PDF expression:

lnL(λ) = ln(λ^3 * x1^2 * e^(-λ*x1)/2) + ln(λ^3 * x2^2 * e^(-λ*x2)/2) + ... + ln(λ^3 * xn^2 * e^(-λ*xn)/2)

Simplifying further:

lnL(λ) = 3ln(λ) + 2ln(x1) - λx1/2 + 3ln(λ) + 2ln(x2) - λx2/2 + ... + 3ln(λ) + 2ln(xn) - λxn/2

Combining like terms:

lnL(λ) = 3nln(λ) + 2(ln(x1) + ln(x2) + ... + ln(xn)) - (λ/2)(x1 + x2 + ... + xn)

To find the maximum likelihood estimate (λ^ML), we differentiate lnL(λ) with respect to λ and solve for λ when the derivative is equal to zero:

d(lnL(λ))/dλ = 3n/λ - (x1 + x2 + ... + xn)/2 = 0

Multiplying through by λ and rearranging:

3n = (x1 + x2 + ... + xn)/2

Simplifying further:

λ^ML = (2n)/(x1 + x2 + ... + xn)

Therefore, the maximum likelihood estimate of λ is equal to (2n)/(x1 + x2 + ... + xn), expressed as a function of n and s where s = x1 + x2 + ... + xn.

To find the maximum likelihood estimate (^λML) of the unknown parameter λ, we need to maximize the likelihood function L(λ), which is a product of the probability density function (PDF) of each observed value.

The PDF for each individual observation Xi is given by:
fX(xi) = (λ^3 * xi^2 * e^(-λxi))/2

For n independent observations, the joint probability density function (PDF) is the product of individual PDFs:
L(λ) = fX(x1) * fX(x2) * ... * fX(xn)

Taking the logarithm of the likelihood function (log-likelihood) helps to simplify the maximization problem without changing the location of the maximum:
log L(λ) = log fX(x1) + log fX(x2) + ... + log fX(xn)

Now, let's find the maximum likelihood estimate (^λML) by maximizing the log-likelihood function.

1. Take the logarithm of the PDF:
log fX(xi) = log[(λ^3 * xi^2 * e^(-λxi))/2]
= log(λ^3) + 2log(xi) - λxi - log(2)

2. Substitute the log PDF into the log-likelihood function:
log L(λ) = log fX(x1) + log fX(x2) + ... + log fX(xn)
= log(λ^3) + 2log(x1) - λx1 - log(2) + log(λ^3) + 2log(x2) - λx2 - log(2) + ... + log(λ^3) + 2log(xn) - λxn - log(2)
= nlog(λ^3) + 2(log(x1) + log(x2) + ... + log(xn)) - λ(s) - nlog(2)

Note: s = x1 + x2 + ... + xn (sum of observed values).

3. Differentiate the log-likelihood function with respect to λ:
d(log L(λ))/dλ = 0 - (s) - 0 + 3n/λ - 0
= 3n/λ - (s)

4. Set the derivative equal to zero and solve for ^λML:
3n/λ - s = 0
3n/λ = s
^λML = 3n/s

Therefore, the maximum likelihood estimate of λ (^λML) is given by ^λML = 3n/s.

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