PUMaC 2019 · 组合（B 组） · 第 5 题

PUMaC 2019 — Combinatorics (Division B) — Problem 5

Marko lives on the origin of the Cartesian plane. Every second, Marko moves 1 unit up with probability 2 / 9, 1 unit right with probability 2 / 9, 1 unit up and 1 unit right with probability 4 / 9, and he doesn’t move with probability 1 / 9. After 2019 seconds, Marko ends up on the point ( A, B ). What is the expected value of A · B ?

解析

Marko lives on the origin of the Cartesian plane. Every second, Marko moves 1 unit up with probability 2 / 9, 1 unit right with probability 2 / 9, 1 unit up and 1 unit right with probability 4 / 9, and he doesn’t move with probability 1 / 9. After 2019 seconds, Marko ends up on the point ( A, B ). What is the expected value of A · B ? Proposed by Alan Yan. Answer: 1811716 . Solution: Define the random variables x , y for 1 ≤ i ≤ 2019 where each x equals 1 if on the i i i i th move, Marko makes a contribution to the right and zero otherwise. y is equal to 1 if on i the ith move we make a contribution upwards and 0 otherwise. Hence, the answer is   n n 2 ∑ ∑ ∑ 4 n   E x · y = E [ x y ] = = 1811716 . i j i j 9 i =1 j =1 i,j Note that one must do cases on whether i = j , but the numbers are such that everything is 4 / 9.