# Definition:Probability Limit

## Definition

Let $T$ be a time series.

Let $S$ denote the range of $T$.

Let $L$ denote the set of lead times of $T$.

Let $\hat z_t$ be a forecast function on $L$.

Let $\map {\hat z_t} l$ denote the forecast value of the observation at the timestamp of lead time $l$.

A probability limit, for a given probability $p$, is the deviation $\Delta_p$, either positive or negative, from $\map {\hat z_t} l$ such that the probability that the actual value lies within $\Delta_p$ of $\map {\hat z_t} l$ is greater than $p$.

That is, the actual value of the time series at $l$, when it occurs, will be within those probability limits within that stated probability $p$.

Typical values for $p$ are whatever may be convenient, for example $50 \%$ or $95 \%$.

### Upper

Let $\Delta_p$ be a probability limit for $\map {\hat z_t} l$.

The upper probability limit of $\map {\hat z_t} l$ is the value $\map {\hat z_t} l + \Delta_p$.

### Lower

Let $\Delta_p$ be a probability limit for $\map {\hat z_t} l$.

The lower probability limit of $\map {\hat z_t} l$ is the value $\map {\hat z_t} l - \Delta_p$.

## Sources

$1$: Introduction:
$1.1$ Four Important Practical Problems:
$1.1.1$ Forecasting Time Series