# 2.11. RV: Random Variables¶

The RV tab allows the user to specify the probabilistic distribution for the random problem at hand. The following probabilistic distributions for the random variables are currently supported:

## 2.11.1. Probability distribution¶

Note

To add a new random variable the user presses the Add button. To remove a random variable, the user must first select it by checking the small circle before the random variable, and then pressing the Remove button.

### 2.11.1.1. Dakota Engine¶

The following six distribution classes are supported for the Dakota UQ engine.

1. Normal (Gaussian)

The user provides the mean ($$\mu$$) and standard deviation ($$\sigma$$) of the normal distribution. The density function of the normal distribution, as a function of $$\mu$$ and $$\sigma$$ is:

(2.11.1.1.1)$f(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left( -{\frac{1}{2} \left( \frac{x - \mu}{\sigma} \right)^2} \right)$
2. Lognormal

The user provides the mean ($$\mu$$) and standard deviation ($$\sigma$$) of the lognormal distribution. The density function of the lognormal distribution, as a function of $$\mu$$ and $$\sigma$$ is:

(2.11.1.1.2)$f(x) = \frac{1}{\sqrt{2 \pi} \zeta x} \exp \left( -{\frac{1}{2} \left( \frac{\ln x - \lambda}{\zeta} \right)^2} \right)$

where $$\zeta^2 = \ln \left( \frac{\sigma^2}{\mu^2} + 1 \right)$$ and $$\lambda = \ln(\mu) - \frac{\zeta^2}{2}$$

3. Beta

The user provides $$\alpha$$, $$\beta$$, lower bound ($$L_B$$), and an upper bound ($$U_B$$) for the beta distribution. The density function of the normal distribution, as a function of these quantities, is:

(2.11.1.1.3)$f(x) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} \frac{(x - L_B)^{\alpha-1}(U_B-x)^{\beta-1}}{(U_B - L_B)^{\alpha + \beta - 1}}$

where $$\Gamma(\alpha)$$ is the Gamma function.

4. Uniform

The user provides the lower bound ($$L_B$$), and an upper bound ($$U_B$$) for the uniform distribution. The density function of the normal distribution, as a function of these quantities, is:

(2.11.1.1.4)$f(x) = \frac{1.0}{(U_B - L_B)}$

The mean of the distribution is $$\frac{(U_B + L_B)}{2.0}$$

5. Weibull

The user provides the shape parameter ($$k$$) and scale parameter ($$a_n$$) for the Weibull distribution. The density function of the Weibull distribution, as a function of these quantities, is:

(2.11.1.1.5)$f(x) = \frac{k}{a_n}\left(\frac{x}{a_n}\right)^{k-1} \exp \left( -(x/a_n)^{k} \right)$

where $$k,a_n > 0$$ and $$x \geq 0$$. For $$x<0$$, $$f(x) = 0$$.

6. Gumbel

The user provides $$\alpha$$ and $$\beta$$ for the Gumbel distribution, where $$\beta$$ is known as the location parameter and $$\frac{1}{\alpha}$$ the scale parameter. The density function of the Gumbel distribution, as a function of these quantities, is:

(2.11.1.1.6)$f(x) = \alpha e^{-\alpha(x-\beta)} \exp(-e^{-\alpha(x-\beta)})$

For each random variable, the user must enter a name and select from the drop-down menu the distribution associated with the random variable. For the distribution selected, the user must then provide the input arguments, which are as described above. Fig. 2.11.1.1.1 shows the panel for a problem with four Random Variables with all random input following Gaussian distributions.

Fig. 2.11.1.1.1 Random variable specification.

Note

To add a new random variable the user presses the Add button. To remove a random variable, the user must first select it by checking the small circle before the random variable, and then pressing the Remove button.

Warning

Removing a random variable may have unintended consequences and cause the UQ engine to fail.

Fig. 2.11.1.1.2 Distributions offered by the R2D app.

Tip

Summary of capabilities and limitations

• o Specify 12 different kinds of random distributions either by parameters, moments (mean and variance), or data samples (Specify 7 different kinds of random distributions by parameters when using Dakota instead of SimCenterUQ)

• o Draw correlated samples through Nataf transformation (Gaussian-copular)

• x Explicitly specify random fields (planned)

• x Specify user-defined random distribution (planned)

• x Specify Non-Gaussian copular correlation (upon request)