# Prediction interval and probability to increase

Let be $$y_i$$ some observed values of a given time-series. We denote $$\hat{y}_i$$ the corresponding predicted values. We also assume to have a prediction interval $$p_i$$ such that $$\hat{y}_i \in p_i = [\hat{y}_i^l , \hat{y}_i^u]$$ with $$1-\alpha = 0.95$$ average coverage.

I am wondering whether one can extract the probability that $$y_{i+1}$$ is higher than $$y_i$$ like that : $$P(y_i \leq y_{i+1} | \alpha) = \frac{\hat{y}_{i+1} - \hat{y}_{i+1}^l}{\hat{y}_{i+1}^u - \hat{y}_{i+1}^l}$$

Can someone explain whether that idea makes sense or not ? I would really appreciate any help, adding some sources would be great :) Many thanks :)