06 Apr 2021

Posted on:

18 Sept 2020

0

# Exponent tends to zero

Hi,   Could you please explain how the exponent tends to zero in calculation part of finding expected value in probability normal distribution.
Instructor
Posted on:

06 Apr 2021

0

Hey Rajesh,

I assume you are refering to the additional materials concerning Normal Distribution in the Probability course.
For future enquiries it is highly advisable to mark the question, pointing out the section which causes you trouble.
That would be of great help to instructors and you would be able to receive answers at shorter notice.
Apart from that, it's wonderful to see keen interest in the derivations of the Normal Distribution formulas. Let's dive in:
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First off, Step 7 suggest splitting into two simple integrals. The first one is the famous Gaussian integral, of which there are
numerous beautiful solutions. You can search it in the Internet or use Wolfram Alpha (further details in the pdf under
Probabilities / Continuos Distributions, check on the resources). Your immediate question is about the second integral and
Step 8. We go about it step by step:
- We integrate from -inf to +inf t * e^(-t^2) dt. To solve this integral, type it in Wolfram Mathematica or, if you are familiar with solving
indefinite integrals, transfer t to dt to obtain integral from -inf to +inf of (1/2) * e^(-t^2) d(t^2) (because d(t^2)/dt = 2t). Next, you want
to have d(-t^2) instead of d(t^2), so you add a minus in front of the integral and write d(-t^2). Following is direct integration and
application of Newton-Leibniz's theorem. That's how you get the result at the end of Step 7.
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Now, you have e^(-t^2). When t tends to +infinity (t -> +inf), then t^2 also tends to +infinity (t^2 -> +inf) and consequently
-t^2 tends to -infinity (because of the negative sign). It's a limit of e^(-t^2) at t tends to infinity - because t^2 will always tend
to +infinity when t tends to either +infinity or -infinity, it is the case that the exponent tends to zero. This answers the question.
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Try writing this in Wolfram Mathematica (https://www.wolframalpha.com/) and verify for yourself.
`int_{-inf}^{inf} e^(-t^2) dt` will get you the value of the Gaussian integral
`lim_{t to +inf} e^(-t^2)` will get you zero, because `lim_{t to +inf} -t^2` gives -infinity.
Similarly, `lim_{t to -inf} e^(-t^2)` is again zero since `lim_{t to -inf} -t^2` is again -inifinity.
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Hope that has helped.
Best regards,
A., The 365 Team