Help with Integral - Solve 1/Sqrt(2*Pi) x Integral of exp^(-1/2*epsilon^2)

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In summary, the integral of exp^(-1/2*epsilon^2) is the Gaussian integral and it equals to sqrt(2*pi). To solve 1/Sqrt(2*Pi) x Integral of exp^(-1/2*epsilon^2), you can first rewrite it as 1/(Sqrt(2*Pi)) x sqrt(2*pi), which simplifies to just 1. The 1/Sqrt(2*Pi) term is a normalization constant in the Gaussian distribution equation and can be solved analytically using the substitution method. It has various real-world applications in physics, engineering, and statistics.
  • #1
Suz84
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Dear All,

I am currently stuck on how to integrate the following:

1/Sqrt(2*Pi) x Integral of exp^(-1/2*epsilon^2).

The above must be integrated with respect to epsilon between two values, -d2 above and -d1 below. I know that you may be able to use the density function of the standard normal, but I am really stuck.

Please help,

Many thanks,

Suz
 
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  • #2
This is the (integral of the) standard normal distribution. Just look up [itex]\Phi(-d_2)-\Phi(-d_1)[/itex].
 
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  • #3
You can also search for a table with values for the error function.

Daniel.
 

1. What is the integral of exp^(-1/2*epsilon^2)?

The integral of exp^(-1/2*epsilon^2) is the Gaussian integral and it equals to sqrt(2*pi).

2. How do I solve 1/Sqrt(2*Pi) x Integral of exp^(-1/2*epsilon^2)?

To solve 1/Sqrt(2*Pi) x Integral of exp^(-1/2*epsilon^2), you can first rewrite it as 1/(Sqrt(2*Pi)) x sqrt(2*pi), which simplifies to just 1. Therefore, the answer is simply 1.

3. What is the significance of the 1/Sqrt(2*Pi) term in the integral?

The 1/Sqrt(2*Pi) term is a normalization constant in the Gaussian distribution equation. It ensures that the total area under the curve of the Gaussian distribution is equal to 1, making it a probability density function.

4. Can this integral be solved analytically?

Yes, the integral of exp^(-1/2*epsilon^2) can be solved analytically by using the substitution method. However, the resulting integral may not have a closed form solution and may need to be approximated using numerical methods.

5. What are some real-world applications of this integral?

The Gaussian integral has various applications in physics, engineering, and statistics. It is used to solve problems related to probability, normal distribution, heat transfer, and quantum mechanics, among others.

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