# 统计与概率代写|STATISTICS AND PROBABILITY II MATH254 University of Liverpool Assignment

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It’s good to hear that it covers both discrete and continuous random variables, as well as important distributions like the geometric, exponential, and normal distributions.

I also appreciate that the module highlights the importance of measure theory and probability as a foundation for this material. This should help students gain a deeper understanding of the underlying concepts and be better prepared to apply them in real-world situations.

Overall, it sounds like a valuable module for students interested in actuarial science, financial mathematics, statistics, and the physical sciences.

Corrupted by their power, the judges running the popular game show America’s Next Top Mathematician have been taking bribes from many of the contestants. Each episode, a given contestant is either allowed to stay on the show or is kicked off.

If the contestant has been bribing the judges she will be allowed to stay with probability 1. If the contestant has not been bribing the judges, she will be allowed to stay with probability $1 / 3$.

Suppose that $1 / 4$ of the contestants have been bribing the judges. The same contestants bribe the judges in both rounds, i.e., if a contestant bribes them in the first round, she bribes them in the second round too (and vice versa).
(a) If you pick a random contestant who was allowed to stay during the first episode, what is the probability that she was bribing the judges?

(a) We first compute $P\left(S_1\right)$ using the law of total probability.
$$P\left(S_1\right)=P\left(S_1 \mid B\right) P(B)+P\left(S_1 \mid H\right) P(H)=1 \cdot \frac{1}{4}+\frac{1}{3} \cdot \frac{3}{4}=\frac{1}{2} .$$
We therefore have (by Bayes’ rule) $P\left(B \mid S_1\right)=P\left(S_1 \mid B\right) \frac{P(B)}{P\left(S_1\right)}=1 \cdot \frac{1 / 4}{1 / 2}=\frac{1}{2}$.

(b) If you pick a random contestant, what is the probability that she is allowed to stay during both of the first two episodes?

(b) Using the tree we have the total probability of $S_2$ is
$$P\left(S_2\right)=\frac{1}{4}+\frac{3}{4} \cdot \frac{1}{3} \cdot \frac{1}{3}=\frac{1}{3}$$

(c) If you pick random contestant who was allowed to stay during the first episode, what is the probability that she gets kicked off during the second episode?

(c) We want to compute $P\left(L_2 \mid S_1\right)=\frac{P\left(L_2 \cap S_1\right)}{P\left(S_1\right)}$.
From the calculation we did in part (a), $P\left(S_1\right)=1 / 2$. For the numerator, we have (see the tree)
$$P\left(L_2 \cap S_1\right)=P\left(L_2 \cap S_1 \mid B\right) P(B)+P\left(L_2 \cap S_1 \mid H\right) P(H)=0 \cdot \frac{1}{3}+\frac{2}{9} \cdot \frac{3}{4}=\frac{1}{6}$$
Therefore $P\left(L_2 \mid S_1\right)=\frac{1 / 6}{1 / 2}=\frac{1}{3}$.