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

• wombat4000
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.
wombat4000

## 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?

## The Attempt at a Solution

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

but can't integrate this.

see error functions

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

Hi wombat! Welcome to PF

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]​

Still don't really understand what's what - this to clarify this is what i was originally referring to

and this is what i got from the error function page

but i need to integrate from $$-\infty$$ to x.

Last edited:
Thanks for your help by the way!

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:

$$\int_{-\infty}^{x}e^{-t^{2}/2}dt$$​

is the same as:

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

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

[size=-2](if you're happy, don't forget to mark thread "solved"!)[/size]​

i still don't get it.

i know that

and

but i don't end up with

should i being using this?

$$\int_{-x}^{\infty}e^{-t^{2}/2}dt\qquad$$ = $$\int_{-x}^{\infty}{e^{-t^{2}}e^{t^{2}/2}}dt\qquad$$

what should $$\phi$$ equal?

?

I think in this case, x should be greater than zero.

## 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.

• Calculus and Beyond Homework Help
Replies
7
Views
956
• Calculus and Beyond Homework Help
Replies
1
Views
650
• Calculus and Beyond Homework Help
Replies
4
Views
252
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
870
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
1K