What is the origin of the standard normal distribution function?

Thread starter coki2000
Start date Apr 11, 2010
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Distribution Normal Normal distribution

In summary, a standard normal distribution is a bell-shaped curve that represents the distribution of a continuous random variable, with a mean of 0 and a standard deviation of 1. Its properties include a total area under the curve of 1, symmetry around the mean, and a mean, median, and mode that are all equal. It is different from a normal distribution in that it has specific values for the mean and standard deviation. The purpose of a standard normal distribution is to standardize data and make comparisons between different distributions. It is commonly used in statistical analyses and data analysis to identify outliers, make predictions, and determine the significance of relationships between variables.

Apr 11, 2010

coki2000

Hello,
Where does the standart normal distribution function

[tex]\Phi (x)=\frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi }}[/tex]

come from?I wonder it.Please explain to me.Thanks for your helps.

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Apr 11, 2010

mathman

Science Advisor

It comes from the central limit theorem, one of the most important theorems in probability theory. Look it up on google.

Related to What is the origin of the standard normal distribution function?

1. What is a standard normal distribution?

A standard normal distribution is a bell-shaped curve that represents the distribution of a continuous random variable. It is also known as a Gaussian distribution or a normal distribution. It is characterized by a mean of 0 and a standard deviation of 1.

2. What are the properties of a standard normal distribution?

The properties of a standard normal distribution include:

The total area under the curve is equal to 1.
The curve is symmetric around the mean, with 50% of the data falling above the mean and 50% falling below.
The mean, median, and mode are all equal.
About 68% of the data falls within 1 standard deviation of the mean, 95% falls within 2 standard deviations, and 99.7% falls within 3 standard deviations.

3. How is a standard normal distribution different from a normal distribution?

A standard normal distribution is a specific type of normal distribution with a mean of 0 and a standard deviation of 1. A normal distribution can have any mean and standard deviation, and is often used to describe real-world data that may not follow a perfect bell-shaped curve.

4. What is the purpose of using a standard normal distribution?

The standard normal distribution is useful because it allows us to standardize data and make comparisons between different distributions. By converting data to z-scores (the number of standard deviations away from the mean), we can compare data from different distributions and make predictions about the likelihood of certain values occurring.

5. How is a standard normal distribution used in statistics and data analysis?

The standard normal distribution is used in many statistical analyses, such as hypothesis testing and confidence intervals. It is also used to calculate probabilities and make predictions about future data. In data analysis, it can help us identify outliers and determine the significance of relationships between variables.