You run a restaurant with N = 10,000 reservations in a day. Each reservation j has:
-
Reserved party size:
Rj
(positive integer)
-
Actual number of people who show up:
Sj
(can be
<
,
=
, or
>
Rj
)
To reduce measurement effort, you only observe (R,S) for a sample of n = 1,000 reservations. The sampling is size-biased:
-
A reservation with reserved size 10 is
twice as likely
to be sampled as a reservation with reserved size 5.
-
More generally, assume the sampling probability is
proportional to reserved size
:
Pr(reservation j is sampled)∝Rj
.
Task
Using the 1,000 sampled pairs (Ri,Si), estimate the total number of people who would show up across all 10,000 reservations:
TS=∑j=110000Sj.
State any assumptions needed, and explain how you would quantify uncertainty (e.g., a confidence interval).