# Method of moments and integration

• semidevil
In summary, the method of moments involves finding the expected value or mean, which is equal to the integral of the probability distribution function. In this case, the probability distribution function is theta*y^(theta-1). This can be simplified to theta*y^theta. The integration is simple and can be done using basic calculus techniques. Therefore, it is important to have a solid understanding of expected values and basic calculus concepts when using the method of moments.

#### 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??

semidevil said:
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.

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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.

## 1. What is the method of moments?

The method of moments is a statistical technique used to estimate the parameters of a probability distribution by equating population moments to sample moments.

## 2. How does the method of moments work?

The method of moments works by equating the theoretical moments of a probability distribution to the empirical moments of a sample. This allows us to estimate the parameters of the distribution and make inferences about the population.

## 3. What are the advantages of using the method of moments?

One advantage of the method of moments is that it is relatively easy to understand and implement. It also does not require a large sample size to obtain accurate estimates, making it useful in situations where data is limited.

## 4. What is integration in the context of the method of moments?

In the context of the method of moments, integration refers to the process of finding the probability distribution function (PDF) or cumulative distribution function (CDF) of a given distribution. This is necessary in order to equate the theoretical moments to the sample moments.

## 5. Can the method of moments be used for any type of probability distribution?

Yes, the method of moments can be used for any type of probability distribution as long as it has a finite number of parameters. However, it may not always provide the most accurate estimates, especially for complex distributions.