Suppose that we modify the reservoir sampling algorithm of...

Suppose that we modify the reservoir sampling algorithm of Exercise $2.18$ so that, when the $k$ th item appears, it replaces the item in memory with probability $1 / 2$. Describe the distribution of the item in memory.

Key Concepts

Reservoir Sampling

Reservoir sampling is an algorithm used to select a random sample (often one item) from a population of unknown size, typically arriving in a data stream. Its key feature is that it allows one to process each element in a single pass, ensuring that each item seen has a chance of being selected, usually with uniform probability if the algorithm is unmodified.

Update Mechanism

In reservoir sampling the algorithm normally replaces the current candidate with a new one according to a rule that maintains the desired distribution. When the algorithm is modified so that each new item replaces the current candidate with a fixed probability (in this case, 1/2), the updating rule—a replacement decision that does not depend on the index of the candidate other than by its arrival order—is fundamentally changed. This new rule affects the survival chance of any item already in the reservoir.

Sampling Distribution

The sampling distribution refers to the probability with which any given item ends up as the final candidate in memory. Under the modified rule, an item that is selected when it first appears must ‘survive’ a series of independent coin-flip decisions (with probability 1/2 for each subsequent item not to replace it) in order to remain at the end. Since later items can replace it with probability 1/2 each time they appear, the likelihood that any given item stays in memory decreases exponentially with the number of items seen afterward.

Exponential (Geometric) Decay

Because each subsequent new item replaces the current reservoir element with probability 1/2, the chance that an item that was selected remains until the end is given by an exponential decay (or geometric sequence) in the number of items that arrive later. For example, if an item is chosen at position i, then it must avoid replacement on each of the n ? i subsequent arrivals, each with probability 1/2, giving its final survival probability as (1/2)^(n?i+1). This creates a non?uniform distribution on the final output that is heavily biased in favor of later-arriving items.

Please give Ace some feedback