This article describes the types of probability distributions applicable to input variables in @LabAix uncertainties-propagating apps.

Uncertainties-propagating apps (for e.g. irr@u) assume that uncertain input variables follow a probability distribution, to be specified by the user, from the apps’ “Monte-Carlo Simulation” tab.

Three types of probability distributions are available from this tab: uniform, normal, and triangular. The selected distribution will apply to *all* input variables marked as uncertain, as shown in the irr@u example below.

⇓ ⇓ ⇓

In this example:

- Two input variables (a cash outflow at year #0, and a cash inflow at year #1) were defined, and marked as uncertain.
- The probability distribution applied to
*both*input variables is a triangular distribution, characterized by a probability density function shaped like a triangle, and defined by three values: the minimum (lower value of the cashflow), the maximum (upper value of the cashflow), and the peak value or “mode” (most likely value of the cashflow). - The cash outflow (in red) varies between -150 k.USD and -50 k.USD. Its most likely value is set to -150 k.USD + 80% × 100 k.USD = -70 k.USD.
- The cash inflow (in blue) varies between +75 k.USD and +225 k.USD. Its most likely value is set to 75 k.USD + 80% × 150 k.USD = 195 k.USD.
- Here, the “mode” or peak value associated with the triangular distribution is set to 80% of the cashflow range, which means that cashflows are distributed very close to their upper limits (-50 k.USD for the cashlow at year #0, and +225 k.USD for the cashlow at year #1).
- This case is representative of an optimistic scenario as the Internal Rate of Return (IRR) associated with these cashflows will be higher than in cases where cashflows are distributed closer to their lower limits.

The uniform, normal, and triangular distributions are used to describe four types of scenarios: neutral, central, pessimistic, and optimistic.

### Uniform distribution (neutral scenario)

Uniform distributions are used to simulate a **neutral scenario**: all input variables, marked as uncertain, follow a uniform distribution where all values are equally likely.

Min

0%

Mid

50%

Max

100%

### Normal distribution (central scenario)

Normal distributions are used to simulate a **central scenario**: all input variables, marked as uncertain, are scattered around a central value following a normal distribution. The standard deviation (σ) around the central value can be set to σ = 5%, 10%, 20% and 40% leading to the samples shown in the figure below.

Min

0%

Mid

50%

Max

100%

### Triangular distribution

Triangular distributions are used to simulate three types of scenarios, depending on the distributions’ “mode”: a **central scenario** (“mode” = 50%), a **pessimistic scenario** (“mode” < 50%), and an **optimistic scenario** (“mode” > 50%).

Min

0%

Mid

50%

Max

100%

•

Central

scenario

“mode” = 50%

•

Pessimistic

scenario

“mode” < 50%

•

Optimistic

scenario

“mode” > 50%

### Input space sampling

Once the probability distribution is defined, a sequence of numerical experiments (Monte-Carlo Experiments) is auto-generated. Each experiment is described by a set of numerical values — one value for each uncertain input variable –. This process is often referred to as Design of Experiments (DoE).

The statistical method, used by @LabAix uncertainties-propagating apps for generating a near-random sample of numerical values, is the Latin Hypercube Sampling (LHS).

The figure above provides an example of LHS’ based input space sampling for two uncertain variables — v1 and v2 — following, both, a triangular distribution (“mode” = 80%).