|
Size: 975
Comment: Initial commit
|
Size: 1829
Comment: Rewrite
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 3: | Line 3: |
| The '''normal distribution''' is a bell-shaped continuous probability distribution that is parameterized to a mean and standard deviation. | The '''normal distribution''' is a bell-shaped continuous probability distribution function that is parameterized to a mean and standard deviation. |
| Line 13: | Line 13: |
| The distribution is bell-shaped. | The distribution is bell-shaped and parameterized to the [[Analysis/Moment|first and second central moments]]. This has useful consequences for estimating the probability that a given value is in the distribution. For example: |
| Line 17: | Line 18: |
A variable distributed this way is notated like ''X ~ N(μ, σ^2^)''. The probability density function is given (and commonly notated) as: {{attachment:norm.svg}} Correspondingly, the cumulative probability function is commonly notated as ''Φ(.)'' rather than ''F(.)''. === Standard Normal Distribution === The '''standard normal distribution''' is specified as ''Z ~ N(0, 1)''. Compare to the calculation for '''Z scores''' that will be compared to the standard normal distribution: ''Z = (x - μ)/σ''. The probability density function simplifies to: {{attachment:stdnorm.svg}}. This is graphed as: {{attachment:stdnormgraph.png||width=300px}} |
|
| Line 22: | Line 47: |
| == Statistics == | == Moments == |
| Line 24: | Line 49: |
| The mean and standard deviation are necessarily given by the distribution formulation. | The [[Analysis/ExpectedValue|expected value]] is given as ''E[X] = μ''. [[Analysis/Variance|Variance]] is given as ''Var[X] = σ^2^''. |
| Line 34: | Line 61: |
| === Standard Normal Distribution === | === Probability Tests === |
| Line 36: | Line 63: |
| The normal distribution characterized by a mean of 0 and a standard deviation of 1 is called the '''standard normal distribution'''. This distribution is referenced for '''Z scores''' (alternatively called '''Z statistics'''). | As noted above, '''Z scores''' (alt. ''Z statistics'') are test statistics for which likelihood can be calculated using the standard normal distribution. |
Normal Distribution
The normal distribution is a bell-shaped continuous probability distribution function that is parameterized to a mean and standard deviation.
Contents
Description
The distribution is bell-shaped and parameterized to the first and second central moments. This has useful consequences for estimating the probability that a given value is in the distribution. For example:
- 68.27% of the cumulative distribution is within 1 standard deviation of the mean
- 95.45% within 2
- 99.73% within 3
A variable distributed this way is notated like X ~ N(μ, σ2).
The probability density function is given (and commonly notated) as:
Correspondingly, the cumulative probability function is commonly notated as Φ(.) rather than F(.).
Standard Normal Distribution
The standard normal distribution is specified as Z ~ N(0, 1).
Compare to the calculation for Z scores that will be compared to the standard normal distribution: Z = (x - μ)/σ.
The probability density function simplifies to:
.
This is graphed as:
Moments
The expected value is given as E[X] = μ.
Variance is given as Var[X] = σ2.
Usage
Probability Tests
As noted above, Z scores (alt. Z statistics) are test statistics for which likelihood can be calculated using the standard normal distribution.
As an example, for a two-tailed test and a significance level of 5%, the critical Z score value is 1.96.