题目
Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is and the third coin is biased so that the probability of obtaining a head is . The random variable is the number of heads that Nikita obtains when he throws all three coins at the same time.
(a) Find the probability generating function of .
Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable is the number of 4s that Rajesh obtains when he throws the two dice.
The random variable is the sum of the number of heads obtained by Nikita and the number of 4s obtained by Rajesh.
(b) Find the probability generating function of , expressing your answer as a polynomial.
(c) Use your answer to part (b) to find .
题目中文翻译
Nikita 有三枚硬币。一枚硬币是公平的，一枚硬币有偏，使得得到正面的概率为 ，第三枚硬币有偏，使得得到正面的概率为 。随机变量 是 Nikita 同时抛掷这三枚硬币时得到的正面数量。
(a) 求 的概率生成函数。
Rajesh 有两个公平的六面骰子，面上标有 1, 2, 3, 4, 5, 6。随机变量 是 Rajesh 抛掷两个骰子时得到 4 的数量。
随机变量 是 Nikita 得到的正面数量与 Rajesh 得到的 4 的数量之和。
(b) 求 的概率生成函数，并将答案表示为多项式。
(c) 使用你在(b)中的答案求 。