Both DLA and PIP are paid 4 weeks in arrears. PIP is generally paid on the same day of the week as the decision to award benefit. If therefore the decision was made on Thursday 20 October then your payment may not come through until this Thursday. I would also expect that this first payment would cover the few days gap between DLA ending and PIP being awarded.
If in any doubt you can contact PIP to check when the payment will be made, the number is Telephone: 0800 917 2222.
I hope this helps