Hello everyone! I'm taking an Mathematical Stat course at university and I came up with this question.
Question can be found via(the question involves hard to type notations, so I thought of uploading it and sharing: drive (dot) proton (dot) me / urls/FVHX8Y08RR#jb7eaDEeff3H
As for this question(the one I've marked in red colour), we need to find a parameter estimation for alpha.
I have completed part (i) and my immediate next approach for solving part (ii) was to find E(x) and E(x^2) and then find estimator for alpha.
However, here I'm facing trouble with finding E(x^2) using integration. I'm getting the answer for integration related to E(x^2) as follows: E(x^2) = [ alpha*((beta)^alpha)/(2-alpha) ]*[((infinity)^(2-alpha)) - ((beta)^(2-alpha))
Obviously alpha cannot be equal to 2 in this case given above.
Also, from part 1, we know E(x) will only hold when alpha>1.
Your help on finding a parameter estimator for alpha is highly appreciated!
To find an estimator for the parameter alpha in part (ii) of your question, you need to determine the expected value of the random variable X^2 (E(X^2)). However, you seem to be encountering difficulty with the integration involved in finding E(X^2) using the given probability density function (PDF).
Let's break down the process step by step:
1. Start with the PDF:
f(x) = (alpha/beta) * (x/beta)^(alpha-1) * exp(-(x/beta)^alpha) for x >= 0
2. To find E(X^2), we need to calculate the integral of x^2 times the PDF, i.e., ∫[0 to ∞](x^2 * f(x)) dx.
3. Substitute the PDF into the integral:
∫[0 to ∞](x^2 * (alpha/beta) * (x/beta)^(alpha-1) * exp(-(x/beta)^alpha)) dx.
4. Simplify:
∫[0 to ∞](alpha/beta) * (x^(alpha+1)/beta^alpha) * exp(-(x/beta)^alpha) dx.
5. Use substitution to simplify the integral. Let u = (x/beta)^alpha, du = (alpha/beta) * (x^(alpha-1)) dx.
This transforms the integral into:
∫[0 to ∞]u^(2/alpha) * exp(-u) du.
6. Notice that this new integral is the definition of the Gamma function Γ(2/alpha + 1).
7. The Gamma function is defined as:
Γ(z) = ∫[0 to ∞] t^(z−1) * exp(-t) dt.
8. Thus, the integral becomes:
∫[0 to ∞]u^(2/alpha) * exp(-u) du = Γ(2/alpha + 1).
9. Finally, E(X^2) becomes:
E(X^2) = (beta^2) * Γ(2/alpha + 1).
From here, you can see that E(X^2) is dependent on the Gamma function. To find an estimator for alpha, you can use methods such as the method of moments or maximum likelihood estimation.
Keep in mind that the estimator may differ based on which estimation method you intend to use. You may need to refer to your course materials or consult with your instructor for further guidance on choosing an appropriate estimator for alpha in relation to the specific estimation method you are utilizing in this question.