Let n be a positive integer. Suppose you have n objects and n buckets. For each object, randomly place it into a bucket, with each of thenoptions being equally likely. Through this random process, some buckets may end up empty, while other buckets may have mul-tiple objects. For each integer k with 0≤ k ≤n, define pn(k) to be the probability that a randomly chosen bucket contains exactly k objects. By definition, pn(0) +pn(1) +pn(2) +. . .+pn(n−1) +pn(n) = 1. Question: As n→ ∞, determine the exact values (or “limits”) for the probabilities pn(0), pn(1), pn(2), pn(3). Clearly show all of your steps and calculations. Use these results to show that when n is sufficiently large, over 98% of the buckets contain atmost three elements. (This explains why Bucket Sort is such an effective sorting algorithm.)
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
Let n be a positive integer. Suppose you have n objects and n buckets. For each object, randomly place it into a bucket, with each of thenoptions being equally likely. Through this random process, some buckets may end up empty, while other buckets may have mul-tiple objects. For each integer k with 0≤ k ≤n, define pn(k) to be the probability that a randomly chosen bucket contains exactly k objects. By definition, pn(0) +pn(1) +pn(2) +. . .+pn(n−1) +pn(n) = 1.
Question:
- As n→ ∞, determine the exact values (or “limits”) for the probabilities pn(0), pn(1), pn(2), pn(3). Clearly show all of your steps and calculations. Use these results to show that when n is sufficiently large, over 98% of the buckets contain atmost three elements. (This explains why Bucket Sort is such an effective sorting algorithm.)
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
