Sure, here's a hypothetical temporal profile for a Purchase-to-Pay process:

```python
{
   ('Prepare Purchase Request', 'Submit Purchase Order'): (5, 2),   # 5 days on average; 2 days standard deviation
   ('Submit Purchase Order', 'Await Supplier Response'): (7, 3),   # 7 days on average; 3 days standard deviation
   ('Await Supplier Response', 'Await Payment Confirmation'): (10, 5), # 10 days on average; 5 days standard deviation
   ('Await Payment Confirmation', 'Payment Received'): (3, 1) # 3 days on average; 1 day standard deviation
}
```

This represents a process where:
- On average, it takes 5 days from preparing a purchase request to submitting it.
- There's a standard deviation of 2 days, meaning the time could be anywhere between 3 and 7 days for individual instances of this process.
- The time from submitting the Purchase Order to awaiting the supplier's response is on average 7 days, with 3 days as a standard deviation.
- There's a 10-day average wait time from awaiting the supplier's response to awaiting payment confirmation, with a standard deviation of 5 days.
- Finally, the time from awaiting payment confirmation to receiving the payment is on average 3 days, with a standard deviation of 1 day.