Method of moments and integration

[semidevil]

i'm learning the method of moments and I'm not understanding on how to do it.

so for example, fy(y, theta) = theta*y^(theta -1). 0<=y<=1.

to find the method of moments estimate for theta.

the book does E(y) = the integeral from 0 to 1 of y * theta*y^(theta-1) dy.

and that becomes theta * y^theta+1/theta + 1

and that becomes theta/ theta + 1...


first question. am I really suppose to integrate y * theta*y^(theta-1) dy?

first of all, the book has never gone to any of the harder integration techniques, and i really have trouble integrating this. how did they do the steps??

 

[juvenal]

i'm learning the method of moments and I'm not understanding on how to do it.

so for example, fy(y, theta) = theta*y^(theta -1). 0<=y<=1.

to find the method of moments estimate for theta.

the book does E(y) = the integeral from 0 to 1 of y * theta*y^(theta-1) dy.

and that becomes theta * y^theta+1/theta + 1

and that becomes theta/ theta + 1...


first question. am I really suppose to integrate y * theta*y^(theta-1) dy?

first of all, the book has never gone to any of the harder integration techniques, and i really have trouble integrating this. how did they do the steps??

Have you learned expected values or means in your class yet? If not - then your class is being badly taught.

The basic formula for calculating the expected value given some pdf is:

Integral[ y f(y) dy]

given a probability distribution function (pdf) f(y). That is equal to the mean, or first moment, and corresponds to the integral you give above.

Second, the integral that you give can be done by someone who knows one month of high-school calculus.

y* theta * y^{theta-1} can be simplified to theta* y^{theta}. Since theta is fixed, the integration is simple. Can you integrate 2*y^2? How about 3*y^3? Same thing.

 

[tacman]

Yes, you are correct in integrating y * theta*y^(theta-1) dy to find the method of moments estimate for theta. This is because the method of moments is a technique used to estimate the parameters of a probability distribution by equating the theoretical moments of the distribution with the observed moments of the data. In this case, the theoretical moment is represented by E(y), which is the expected value of y, and is found by integrating y * theta*y^(theta-1) dy over the range of y from 0 to 1.

As for the integration technique, it may seem difficult at first, but with practice and understanding of basic integration principles, it can become easier. In this case, you are integrating a power function, which follows the rule of integration by substitution. You can use the substitution u = theta * y^(theta-1) to simplify the integration. The steps may vary depending on your specific textbook or course material, but the general method is to substitute and then integrate using the power rule. I recommend practicing with simpler integration problems before attempting this one to build your understanding and confidence. You can also seek help from your instructor or classmates for further explanation and practice.