How can the integral of a normal distribution be solved using substitution?

In summary, the conversation discusses difficulty integrating a function on the normal distribution and a possible solution using the substitution method. The use of error functions is also suggested.
  • #1
wombat4000
36
0

Homework Statement



I'm having difficulty integrating something,
click http://en.wikipedia.org/wiki/Normal_distribution
and under Cumulative distribution function, there is an integral - how do you get to the next line?


Homework Equations





The Attempt at a Solution




i have tried the substitution of [tex]t=u-\mu[/tex] which gives
[tex]\int_{-\infty}^{x}e^{-t^{2}/2}dt[/tex]

but can't integrate this.
 
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  • #2
see error functions

wombat4000 said:
i have tried the substitution of [tex]t=u-\mu[/tex] which gives
[tex]\int_{-\infty}^{x}e^{-t^{2}/2}dt[/tex]
but can't integrate this.

Hi wombat! Welcome to PF :smile:

All you need is in http://en.wikipedia.org/wiki/Error_function

Good luck!

[size=-2](if you're happy, don't forget to mark thread "solved"!)[/size]​
 
  • #3
Still don't really understand what's what - this to clarify this is what i was originally referring to
f26a8934f9cfafa33024e7ade8201463.png

and this is what i got from the error function page
7777288be8057a4c26904e48e168915c.png

but i need to integrate from [tex]-\infty[/tex] to x.
 
Last edited:
  • #4
Thanks for your help by the way!
 
  • #5
An even function!

Ah! You didn't take account of the fact that the integrand is the same for -t as for t (an "even function"), so you can mutiply the limits by -1:

[tex]\int_{-\infty}^{x}e^{-t^{2}/2}dt[/tex]​

is the same as:

[tex]\int_{-x}^{\infty}e^{-t^{2}/2}dt\qquad,[/tex]​

which is (a multiple of) erfc(-x).

Then the top of the wikipedia page gives you the answer! :smile:

[size=-2](if you're happy, don't forget to mark thread "solved"!)[/size]​
 
  • #6
i still don't get it.
 
  • #7
i know that
4ecb9cb151969361ae1b2357fea5d66f.png

and
066ed14d31d7ea03d453237b18111eba.png

but i don't end up with
f26a8934f9cfafa33024e7ade8201463.png
 
  • #8
should i being using this?

[tex]\int_{-x}^{\infty}e^{-t^{2}/2}dt\qquad[/tex] = [tex]\int_{-x}^{\infty}{e^{-t^{2}}e^{t^{2}/2}}dt\qquad[/tex]
 
  • #9
what should [tex]\phi[/tex] equal?

003dabb870f6a1fc0521a85000ea8090.png
?
 
  • #10
I think in this case, x should be greater than zero.
 

Related to How can the integral of a normal distribution be solved using substitution?

1. What is the normal distribution integral?

The normal distribution integral, also known as the cumulative distribution function, is a mathematical function used to calculate the probability of a random variable falling within a certain range of values in a normal distribution. It is represented by the area under the normal curve and ranges from 0 to 1.

2. How is the normal distribution integral calculated?

The normal distribution integral is calculated by integrating the probability density function (PDF) of a normal distribution over a given range of values. This involves using a mathematical formula and can also be approximated using statistical software or tables.

3. What is the significance of the normal distribution integral?

The normal distribution integral is significant because it allows us to calculate the probability of obtaining a specific value or range of values in a normal distribution. This is important in many fields, including statistics, economics, and social sciences, as it helps us understand and make predictions about real-world phenomena.

4. How does the normal distribution integral relate to the normal distribution curve?

The normal distribution integral is represented by the area under the normal distribution curve. The curve is bell-shaped and symmetrical, with the mean, median, and mode all equaling the same value. The integral allows us to calculate the probability of obtaining a certain value or range of values in the normal distribution.

5. Can the normal distribution integral be used for non-normal distributions?

The normal distribution integral is specifically designed for normal distributions. It cannot be used for non-normal distributions, as it assumes that the data follows a specific pattern. However, there are other types of distribution integrals that can be used for non-normal distributions, such as the uniform distribution integral or the exponential distribution integral.

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