Continuous Distributions

A continuous distribution describes the probabilities of the possible values of a continuous random variable. A continuous random variable is a random variable with a set of possible values (known as the range) that is infinite and uncountable. 

All user-defined distributions used in HEC-FDA are continuous distributions (result collection and processing rely on empirical distributions and histograms). Four different forms of continuous distributions can be selected in HEC-FDA: 1) uniform; 2) triangular; 3) normal; and, 4) log normal.

Any of the continuous distributions can be defined for a number of parameters and inputs to HEC-FDA. A list of the parameters and inputs for which a continuous distribution can be specified in HEC-FDA is shown listed:

• Structure, Content, Other, and Vehicle Depth-Percent Damage Functions
• Foundation Height
• Structure Value as percent of Mean
• Content Value as percent of Mean
• Content to Structure Value Ratio (percent)
• Other to Structure Value Ratio (percent)
• Vehicle to Structure Value Ratio (percent)

The four different forms of continuous distributions are described in the following sections

Distributions

Log Normal

A variable x is log-normally distributed if ln(x) is normally distributed. Log normal distribution is often used to assist in parameterizing random variables that are always greater than zero. The PDF of the log normal distribution is described mathematically in terms of µ and σ and is described in the equation below. The log Normal distribution is used as a special case of the Log Pearson Type III distribution when the skew of the analytical flow-frequency function is 0.

$//<![CDATA[ \begin{array}{l}f(x)=\frac{1}{\sigma\sqrt{2\pi}}e_{}^{-\frac{\left(\left(\ln\left(x\right)-\mu\right)^2)\right)}{2\sigma^2}}\end{array} //]]>$

The CDF of the log normal distribution is described mathematically in the following equation:

$//<![CDATA[ \begin{array}{l}F(x)=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{ln(x)-\mu}{\sigma{\sqrt{2}}}\right)\right]\end{array} //]]>$

The inverse CDF is a simple transformation of the normal distribution CDF and is shown in the following equation:

$//<![CDATA[ \begin{array}{l}F^{-1}(p)=e^{{\mu}+{\sigma}*Z(p)}\end{array} //]]>$

Normal

The normal distribution is one of the most commonly used distributions in evaluating uncertainty. HEC-FDA requires that the mean (µ) and the standard deviation (σ) of the distribution be specified. In a normal distribution, the value range is always negative infinity to positive infinity, so it is important to consider the repercussions of choosing this distribution. The PDF of a normal distribution is displayed in the figure below, and is described mathematically in terms of µ and σ in the equation below.

Probability Density Function (PDF) of a Normal Distribution

$//<![CDATA[ \begin{array}{l}f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\end{array} //]]>$

The CDF of a normal distribution is displayed in the figure below and described mathematically in the equation below.

Cumulative Distribution Function (CDF) of a Normal Distribution

$//<![CDATA[ \begin{array}{l}F(x)=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\left(x-\mu\right)}{\sigma{\sqrt{2}}}\right)\right]\end{array} //]]>$

The error function, denoted as erf(x) in the above equation, is an integral that cannot be expressed in elementary functions and can only be approximated. A two-parameter probability distribution defined by the mean and standard deviation. A symmetrical “bell shaped” curve applicable to many kinds of data sets where values are equally likely to be greater than or less than the mean. Also called a Gaussian distribution.  The distribution is truncated at three standard deviations.

The t value is only evaluated from zero to 0.5, so if p is greater than 0.5, the value 1-p is evaluated in t. This assumption is appropriate given that the normal distribution is symmetrical. The inverse CDF for a normal distribution is shown in the equation below.

$//<![CDATA[ \begin{array}{l}F^{-1}(p)=\left[\begin{matrix}(a*\sigma)+Mean & for\rho < 0.5 \\ Mean & for \rho = 0.5\\ (-a*\sigma)+ Mean & for \rho >0.5 \end{matrix}\right]\end{array} //]]>$

Where a is defined in the following equation as:

$//<![CDATA[ \begin{array}{l}a=t+\frac{(c0+c1*t+c2*t^2)}{(1+d1*t+d2*t^2+d3*t^3)}\end{array} //]]>$

where:
c0 = 2.515517
c1 = 0.802853
c2 = 0.010328
d1 = 1.432788
d2 = 0.189269
d3 = 0.001308

$//<![CDATA[ \begin{array}{l}t=(\sqrt{ln(1/p2}))\end{array} //]]>$

The standard normal distribution, where mean (µ) = 0 and standard deviation (σ) = 1, is generally expressed as Z as shown in the equation below.

$//<![CDATA[ \begin{array}{l}Z=\sqrt{n}(\frac{1}{n}\sum_{i=1}^{n}x_{i})=\sqrt{n}E(X)=\sqrt{nX}=\sqrt{n}\frac{\overline{X}-\mu}{\sigma}=\frac{\overline{X}-\mu}{\sigma/\sqrt{n}}\end{array} //]]>$

Triangular

A triangular distribution requires that the minimum, maximum, and mode values of the parameter of interest be entered into HEC-FDA. Note that the mode value may or may not be the mean of the distribution and must fall between the minimum and maximum values. The PDF of a triangular distribution is shown in the figure below. The PDF is described mathematically in the equation below, with the assumption that a ≤ c ≤ b.

Probability Density Function (PDF) of a Triangular Distribution

$//<![CDATA[ \begin{array}{l}f(x)= \begin{cases}0 & \text { for } x< a \\ \frac{2(x-a)}{(c-a)(b-a)} & \text { for } a \leq x \leq c \\ 1-\frac{2(c-x)}{(c-a)(c-b)} & \text { for } c< x \leq b \\ 0 & \text { for } b< x\end{cases}\end{array} //]]>$

The Cumulative Distribution Function (CDF) of a triangular distribution is shown in the figure below (assuming that a ≤ c ≤ b.), and is described mathematically in the equation below.

Cumulative Distribution Function (CDF) of a Triangular Distribution

$//<![CDATA[ \begin{array}{l}f(x)= \begin{cases}0 & \text { for } x< a \\ \frac{(x-a)^2}{(c-a)(b-a)} & \text { for } a \leq x \leq c \\ 1-\frac{(c-x)^2}{(c-a)(c-b)} & \text { for } c< x \leq b \\ 1 & \text { for } b< x\end{cases}\end{array} //]]>$

The inverse cumulative distribution for a triangular distribution, given a ≤ c ≤ b, is as follows:

$//<![CDATA[ \begin{array}{l}F^{-1}(p)= \begin{cases}a+\sqrt{(p)(c-a)(b-a)} & \text { for } 0 \leq p \leq \frac{b-a}{c-a} \\ c-\sqrt{(1-p)(c-a)(c-b)} & \text { for } \frac{b-a}{c-a}< p< 1\end{cases}\end{array} //]]>$

Uniform

The uniform distribution is the most basic of the four distributions. The two required input parameters for the uniform distribution are the maximum and minimum possible values for the parameter of interest. The maximum and minimum values may be any real number. The Probability Density Function (PDF) of a uniform distribution is shown in the figure below. The PDF is described mathematically in the equation below, where a is the minimum and b is the maximum value of x.

Probability Density Function (PDF) of a Uniform Distribution

$//<![CDATA[ \begin{array}{l}f(x)= \begin{cases}\frac{1}{b-a} & \text { for } a \leq x \leq b \\ 0 & \text { for } x< a \text { or } x>b\end{cases}\end{array} //]]>$

The Cumulative Distribution Function (CDF) of a uniform distribution is shown in the figure below. The CDF is described mathematically in the equation below:

Cumulative Distribution Function (CDF) of a Uniform Distribution

$//<![CDATA[ \begin{array}{l}F(x)=\int_{-\infty}^{\infty} f(x) d x= \begin{cases}0 & \text { for } x< a \\ \frac{x-a}{b-a} & \text { for } a \leq x< b \\ 1 & \text { for } x \geq b\end{cases}\end{array} //]]>$

Solving for x in the solution space of (0,1], the inverse CDF for the continuous uniform distribution is shown in the equation below:

$//<![CDATA[ \begin{array}{l}\left.\mathrm{F}^{-1}(\mathrm{p})=\mathrm{a}+(\mathrm{b}-\mathrm{a}) * \mathrm{p}\right) \mid 0 \leq \mathrm{p}< 1\end{array} //]]>$