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Continuous probability distributions

Normal distribution which is also called the Gaussian distribution or a bell curve, is one of the most important distribution in statistics. It has very interesting mathematical features and occurs very often in nature. It is usually designated with $N(\mu,\sigma)$ .

A density function is defined by: $\begin{displaymath} f(x,\mu,\sigma)=\frac{1}{\sqrt{2\pi}\sigma}\exp\bigg(-\frac{(x-\mu)^2}{2\sigma^2}\bigg), \label{r_normalny_fun} \end{displaymath}$

where:

$-\infty<x<+\infty$ ,

$\mu$ – an expected value of population (its measure is mean),

$\sigma$ – standard deviation.

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Normal distribution is a symmetrical distribution for a perpendicular line to axis of abscissae going through the points designating the mean, mode and median.

Normal distribution with a mean of $\mu=0$ and $\sigma=1$ ( $N(0,1)$ ), is so called a standardised normal distribution.

t-Student distribution – the shape of t-Student distribution is similar to standardised normal distribution, but its tails are longer. The higher the number of degrees of freedom ( $df$ ), the more similar the shape of t-Student distribution to normal distribution.

A density function is defined by: $\begin{displaymath} f(x,df)=\frac{\Gamma(\frac{df+1}{2})}{\Gamma(\frac{df}{2})\sqrt{df\pi}}\left(1+\frac{x^2}{df}\right)^{-\frac{df+1}{2}}, \end{displaymath}$

where:

$-\infty<x<+\infty$ ,

$df$ – degrees of freedom (sample size is decreased by the number of limitations in given calculations),

$\Gamma$ is a Gamma function.

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Chi-square distribution, this is a right-skewed distribution with a shape depending on the number of degrees of freedom $df$ . The higher the number of degrees of freedom, the more similar the shape of $\chi^2$ distribution to the normal distribution.

Density function is defined by: $\begin{displaymath} f(x,df)=\frac{1}{2^{\frac{df}{2}}\Gamma{\frac{df}{2}}}x^{\frac{df}{2}-1}e^{-\frac{x}{2}}, \end{displaymath}$

where:

$x>0$ ,

$df$ – degrees of freedom (sample size is decreased by the number of limitations in given calculations),

$\Gamma$ is a Gamma function.

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Fisher-Snedecor distribution, this is a distribution which has a right tail that is longer and a shape that depends on the number of degrees of freedom $df_1$ and $df_2$ .

A density function is defined by: $\begin{displaymath} F(x,df_1,df_2)=\frac{\sqrt{\frac{(df_1x)^{df_1}d_2^{df_2}}{(df_1x+df_2)^{df_1+df_2}}}}{xB\left(\frac{df_1}{2},\frac{df_2}{2}\right)}, \end{displaymath}$

where:

$x>0$ ,

$df_1$ , $df_1$ – degrees of freedom (it is assumed that if $X$ i $Y$ are independent with a $\chi^2$ distribution with adequately $df_1$ and $df_2$ degrees of freedom, than $F=\frac{X/df_1}{Y/df_2}$ has a F Snedecor distribution $F(df_1,df_2)$ ),

$B$ is a Beta function.

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