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fX(x)=λ3x2e−λx2, 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
can you explain your answer?
Oh, the mysterious parameter λ, trying to hide from us! Well, let's see if we can find it using our maximum likelihood detective skills.
First, let's write down the likelihood function. Since the random variables are independent, the likelihood function is the product of the individual PDFs:
L(λ) = fX1(x1) * fX2(x2) * ... * fXn(xn)
Plugging in the Erlang PDF, we have:
L(λ) = λ^(3n) * x1^2 * e^(-λx1^2) * x2^2 * e^(-λx2^2) * ... * xn^2 * e^(-λxn^2)
To make things a bit easier, let's work with the log-likelihood function. Taking the natural logarithm of both sides, we get:
ln L(λ) = 3n ln λ + 2Σln xi - λΣxi^2
To find the maximum likelihood estimate (MLE) of λ, we want to find the value that maximizes the log-likelihood function. Taking the derivative with respect to λ and setting it to zero, we have:
d/dλ (ln L(λ)) = 0
3n/λ - Σxi^2 = 0
Solving for λ, we get:
λ^ML = 3n / Σxi^2
Now, we were asked to express our answer as a function of n and s where s = x1 + x2 + ... + xn. Since Σxi^2 can be written as (x1^2 + x2^2 + ... + xn^2), we can substitute s into our equation:
λ^ML = 3n / s
So, the maximum likelihood estimate of λ is 3n divided by the sum of the observed values, s. Keep in mind that this is just an estimate, so it's not a guarantee that it's exactly the true value. Lambda might still be hiding somewhere, but we did our best!
To find the maximum likelihood estimate (MLE) of λ based on the observed values x1, x2, ..., xn, we need to maximize the likelihood function.
The likelihood function is given by the product of the probability density functions (PDFs), since the random variables are independent:
L(λ) = fX1(x1) * fX2(x2) * ... * fXn(xn)
= λ^(3n) * (x1^2 * e^(-λx1^2)) * (x2^2 * e^(-λx2^2)) * ... * (xn^2 * e^(-λxn^2))
To simplify the likelihood function, we can take the logarithm of both sides:
ln(L(λ)) = ln(λ^(3n)) + ln(x1^2 * e^(-λx1^2)) + ln(x2^2 * e^(-λx2^2)) + ... + ln(xn^2 * e^(-λxn^2))
= 3n * ln(λ) + 2∑ln(xi) - λ∑xi^2
Now, we want to find the value of λ that maximizes this likelihood function. This is equivalent to finding the value of λ that minimizes the negative logarithm of the likelihood function. So we have:
-ln(L(λ)) = -3n * ln(λ) - 2∑ln(xi) + λ∑xi^2
Now, we differentiate with respect to λ and set the derivative equal to zero to find the critical point:
d/dλ (-ln(L(λ))) = -3n/λ + ∑xi^2 = 0
Simplifying further:
3n/λ = ∑xi^2
Now, solving for λ:
λ = 3n/∑xi^2
Since s = x1 + x2 + ... + xn, we can substitute ∑xi^2 with s:
λ = 3n/s
Therefore, the maximum likelihood estimate of λ, λ^ML, is given by λ^ML = 3n/s.
To find the maximum likelihood estimate of λ based on observed values x1, x2, ..., xn, we can use the principle of maximum likelihood estimation.
The likelihood function for the observed values x1, x2, ..., xn is given by the product of the probability density function (PDF) values corresponding to each observation. Since the random variables X1, X2, ..., Xn are independent, the likelihood function is their joint PDF:
L(λ) = fX1(x1) * fX2(x2) * ... * fXn(xn)
Substituting the Erlang PDF fX(x) = λ^3x^2e^(-λx^2) into the likelihood function, we get:
L(λ) = λ^3(x1^2)(e^(-λx1^2)) * λ^3(x2^2)(e^(-λx2^2)) * ... * λ^3(xn^2)(e^(-λxn^2))
To simplify this expression, we can take the logarithm of the likelihood function (log-likelihood) since the logarithm is a monotonically increasing function. This helps in computation and has no effect on finding the maximum point:
log L(λ) = 3log λ + 2log(x1) - λx1^2 + 2log(x2) - λx2^2 + ... + 2log(xn) - λxn^2
To find the maximum likelihood estimate of λ, we differentiate the log-likelihood function with respect to λ, set it to zero, and solve for λ:
d(log L(λ))/dλ = 0
Differentiating the log-likelihood function, we get:
3/λ - 2x1^2 - 2x2^2 - ... - 2xn^2 = 0
Rearranging the terms, we get:
3/λ = 2(x1^2 + x2^2 + ... + xn^2)
Multiplying both sides by λ, we have:
3 = 2λ(x1^2 + x2^2 + ... + xn^2)
Finally, solving for λ, we find:
λ^ML = 3 /[2(x1^2 + x2^2 + ... + xn^2)]
Expressing the answer as a function of n and s, where s = x1 + x2 + ... + xn:
λ^ML = 3 /[2(s^2)]