Three Delay Classes, One Nudge

A food-delivery app predicts how late each order will be. Its model sorts the order into one of three delay classes, then reports the expected delay: each class's delay weighted by its probability.

The final layer emits three logits, softmax turns them into probabilities, and the expected delay is computed from those:

$$p_k = \frac{e^{z_k}}{e^{z_1} + e^{z_2} + e^{z_3}}, \qquad \hat{y} = 0\,p_1 + 10\,p_2 + 40\,p_3$$

For one order the logits are

$$(z_1,\ z_2,\ z_3) = (1.2,\ 0.8,\ 0.5)$$

The backward pass needs the gradient of $\hat{y}$ with respect to every logit. Find one of them: the rate of change of the expected delay with respect to the "a bit late" logit,

$$\frac{\partial \hat{y}}{\partial z_2}$$

at these logits, in minutes per unit of logit.

Give your answer rounded to 3 decimal places, including its sign. Carry at least four decimals through the probabilities.