# What substitution/manipulation was done here?

• MathewsMD
In summary, there is a question regarding the calculation of the mean using the pdf and integral. The conversation discusses the use of mu as a constant and the integration of three terms. It is mentioned that the integral is the same for each value of mu.
MathewsMD
I've attached the image with this post. It looks fairly simply to go from the first line to the second line and almost like a uv substitution, but I tried plugging it in and playing around with the integral and cannot get the exact answer. I really feel like something's just going over my head or that I'm missing a step. Regardless, any help would be great!

Note: mu is the mean.

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MathewsMD said:
I've attached the image with this post. It looks fairly simply to go from the first line to the second line and almost like a uv substitution, but I tried plugging it in and playing around with the integral and cannot get the exact answer. I really feel like something's just going over my head or that I'm missing a step. Regardless, any help would be great!

Note: mu is the mean.

If ##\mu## is the mean, and ##f## is the pdf, then isn't ##\mu = \int xf(x)dx##?

Expand $(x-\mu)^2 = x^2-2x\mu+\mu^2$ to get three integrals and use the definition of $\mu$ to get the second term.

By infinity limits the integral is the same for each value of ##\mu##. Are you post all informations?

Last edited:
$\mu$ is a constant. Therefore $\int_{-\infty}^{\infty}\mu^2f(x)dx=\mu^2\int_{-\infty}^{\infty}f(x)dx=\mu^2$.

Also $\int_{-\infty}^{\infty}\mu xf(x)dx=\mu\int_{-\infty}^{\infty}xf(x)dx=\mu^2$.

## What substitution/manipulation was done here?

1. What type of substitution or manipulation was used in this experiment?A: The specific type of substitution or manipulation will depend on the experiment and the variables being studied. Some common types include control groups, randomization, and blinding.

2. How was the data manipulated or substituted in this study?A: The data may have been manipulated through statistical analysis, such as mean comparisons or regression, or through physical manipulation, such as changing the environment or introducing a treatment.

3. Was the substitution or manipulation ethical?A: Ethical considerations are important in any scientific study. Researchers must follow ethical guidelines and obtain informed consent from participants before manipulating or substituting any variables in their study.

4. What was the purpose of the substitution or manipulation?A: The purpose of the substitution or manipulation in a study is often to test a hypothesis or determine the effect of a specific variable on the outcome of interest.

5. How was the substitution or manipulation controlled for?A: To ensure the validity and accuracy of the results, scientists must control for any potential confounding variables that could affect the outcome. This may involve using control groups or statistical techniques to account for these variables.

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