Probability density function#

• let $x$ be a continous (not discrete!) variable that takes values in $\mathrm{\Omega }$, for example temperature, probability density function $f_X(x)$ of an event X (i.e. T=10°C) is defined as a continous function on $\mathbb{R}$ with the following three attributes:

• Question: What is the unit of the pdf?
  Answer: [$f_X(x)$] = [x]${}^{-1}$.

• Question: What is the integral of the pdf? Answer: The cumulative distribution function.

Cumulative distribution function#

• Cumulative distribution function for an event X is a montonously increasing, non-dimensional function F_X(x) on $\mathbb{R}$ defined as:

• which is equivalent to:

• consequently the probability of the event $X$ to be inside the range of $(a,b)$ is:

Expectation $\epsilon$#

• the expectation of a given pdf weighs is with $x$ in the integral:

• two attributes of the expectation are:

Central moments $\mu$#

• k-th moment of a continous random variable X:

• k-th central moment of a continous random variable X:

• example: anomalies with $\mu$ mean seasonal cycle

• mean $\mu$: location parameter $\mu ={\mu }^{(1)}$

• variance:

• standard deviation:

• Chebyshev’s inequality:

Skewness ${\gamma }_1$#

• is a measure of the asymmetry of a distribution: symmetric for ${\gamma }_1=0$, scaled version of the third central moment, non-dimensional shape parameter

Kurtosis ${\gamma }_2$#

• is a measure of the peakedness of a distribution: a normal distribution (will be explained later this lecture) has ${\gamma }_2=0$, scaled and shifted version of the fourth central moment, non-dimensional shape parameter

Examples#

• summer sea level at Kieler Förde, $\mu =0.06$, $\sigma =0.19$, ${\gamma }_1=-0.6$, ${\gamma }_2=4.07$

• probability densities of some measured variables

P-quantiles#

• mean and variance are affected by the tail ends of the pdf (likelihood of extreme values), but p-quantiles $x_p$ are insensitive to extreme values.

• p quantile of 0.3 means that 30% of the x values are below this threshold

• median m${}_{50}$ is the 50%-quantile: half of the distribution lays above and the other half below m${}_{50}$.

• let’s look at the p-quantiles of the log-normal distribution in Figure 5 to get an idea. note the difference of mean and median!

Uniform distribution#

• symmetric and less peaked than the normal distribution:

• with the cumulative distribution function:

• exercise: calculate $\mu ,Var,\sigma ,{\gamma }_1,{\gamma }_2$ of the uniform distribution $\mathcal{U}\mathcal{(}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{)}$
  solutions: $\mu (\mathcal{U}\mathcal{(}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{)}\mathcal{)}\mathcal{=}\frac{\mathcal{1}}{\mathcal{2}}\mathcal{(}\mathcal{a}\mathcal{+}\mathcal{b}\mathcal{)}$, $Var(\mathcal{U}\mathcal{(}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{)}\mathcal{)}\mathcal{=}\frac{\mathcal{1}}{\mathcal{12}}\mathcal{(}\mathcal{b}\mathcal{-}\mathcal{a}{\mathcal{)}}^{\mathcal{2}}$, $\sigma (\mathcal{U}\mathcal{(}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{)}\mathcal{)}\mathcal{=}\sqrt{\frac{\mathcal{1}}{\mathcal{12}}}\mathcal{(}\mathcal{b}\mathcal{-}\mathcal{a}\mathcal{)}$, ${\gamma }_1(\mathcal{U}\mathcal{(}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{)}\mathcal{)}\mathcal{=}\mathcal{0}$, ${\gamma }_2(\mathcal{U}\mathcal{(}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{)}\mathcal{)}\mathcal{=}\mathcal{-}\mathcal{1.2}$

Normal (Gaussian) distribution#

• most physical quantities are nearly normal distributed

• no skewness or kurtosis: ${\gamma }_1={\gamma }_2=0$

• no analytical form of cdf, approximation:

• central limit theorem states: If $X_k,k=1,2,...$ is an infinite series of independent and identically distributed random variables with $\epsilon (X_k)=\mu$ and $Var(X_k)={\sigma }^2$ then the average $\frac{1}{n}\sum _{k=1}^n{X}_k$ is asymptotically normal distributed. That is:

• a larger sample size reduces the standard deviation as of:

Log-normal distribution#

• distribution of positive definite quantities such as rainfall, wind speed

• with the median value $\theta$ and

• exercise: derive a general for the k-th central moment of the distribution
  solution: $\epsilon (X^k)={\theta }^k{e}^{{k\sigma }^2/2}$

${\chi }^2$-distribution#

• sum of k independent squared $\mathcal{N}\mathcal{(}\mathcal{0}\mathcal{,}\mathcal{1}\mathcal{)}$ random variables, k number of degrees of freedom, application for the pdfs of variance estimates:

• with

• it has handy attributes:

Student’s t-distribution#

• application for testing the significance of the differences in the means. be $t(k)$ a test variable with $k>0$, if A and B are independent random variables such that

• the t-distribution can be written as:

• using the $\mathrm{\Gamma }$-function (28) the distribution can also be written as:

• t-test?

Fisher-F-distribution#

• application for testing the significance of the differences in the variance. for ${\chi }^2$-distributed $K$ and $L$:

• the F-distribution is given by:

• alternatively the probsbility density of the F-distribution is also given by:

Summary of theoretical distributions#

Continous random vectors, multi-variate data#

• example: vectors X temperature and Y sea level pressure: