Since 1950s IRT has been used in order to test educational environment. Here we are output P(a) (the probability that the user will answer the item correctly) given :

1. j - the item
2. $\theta$ - student ability level
3. $\beta_j$ - item difficulty level

This can be formulated as (when sigmoid is used as an item response function):

$P(\alpha) = \sigma (\theta - \beta) = \frac {1} {1 + exp(-\theta -\beta_j)}$

IRTs can also be used to predict $\theta$ or $\beta_j$.