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Normal/Gaussian distribution


  
Figure: An example of a Gaussian distribution curve. The vertical line marks the mean value and the horizontal line shows the $\pm \sigma $ range. The empirical probability distribution for the Bergen September temperature is also shown as black dots. [stats_uib_3_2.m]
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 \begin{displaymath}f(x)= \frac{1}{\sigma \sqrt{2 \pi}} exp \left[ - \frac{(x-\mu)^2}{2 \sigma^2} \right] .
\end{displaymath} (3.4)

There are a number of commonly used theoretical distribution functions, which have been derived for ideal conditions. One such case is where the process ( yi, i=[1...N]) is random (stochastic), and whose distribution follows a Gaussian shape described by f(x) in equation 3.4. This distribution function is widely used in statistical sciences. $\sigma$ is in this case estimated by taking the standard deviation: $\sigma = std(y_i)$, and $\mu$ is taken as the mean value of yi.

Fig. 3.2 shows a typical example of a Gaussian distribution. The values for $\sigma$ and $\mu$ have been taken from the Bergen September 2-m temperature 1861-1997 record, and the empirical histogram for the temperature record is also shown as black dots.

The Gaussian distribution function in Fig 3.2 gives a concise and approximate description of the Bergen September temperature range and likelihood of occurrence. The mean and standard deviation, the two parameters used for fitting the Gaussian function to the observations, give a good description of the Bergen temperature statistics.

Gaussian distribution is also commonly referred to as 'normal distribution'. Continuous distribution.

Central limit theorem: as the sample size of a set of independent observations becomes large, the sum will have a Gaussian distribution.


next up previous contents
Next: Binomial distribution Up: Probability Distributions Previous: Cumulative probability and probability
David Stephenson
2000-09-02