1 . Answer the following questions in technical or non-technical language.

(a) What is meant by the monotonicity of probability? Give a practical example.

(b) What is the principle of inclusion-exclusion? Give a practical example.

(c) What is Bayes' Theorem? Give a practical example.

(d) How do continuous and discrete probability distributions differ?

(e) What does it mean to standardize a random variable?

2 . 210 countries are set to compete in the 2022 World Cup qualifying tournaments which will determine 31 countries who will play in the World Cup (the host country, Qatar, automatically gets one spot).

Write the name of the discrete distribution and write the PMF for the random variables below.

(a) Whether or not a randomly selected team will make it to the World Cup.

(b) Number of times you must randomly choose a country before finding a qualifying team.

(c) Number of qualifying teams in a random sample of 20 countries.

3 . Suppose a random variable X has the pdf f(x)=cx2 on its support [0, 3]. Find the probability (X < 1) and the variance of X.