What is the probability that team A wins the series? Give an exact result and confirm it via simulation.
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Description
Homework 3 FTEC 6321 Due September 16, 2020
1. Suppose 2 teams A and B are playing a series of games and
the first team to win 4 games wins the series. Suppose that team A has a 55%
chance of winning each game and that the outcome of each game is independent.
Hint: binomial distribution
(a) What is the probability that team A wins the series?
Give an exact result and confirm it via simulation.
(b) What is the expected number of games played? Give an
exact result and confirm it via simulation.
(c) What is the expected number of games played given that
team A wins the series? Give an exact result and confirm it via simulation.
(d) Now suppose we only know that team A is more likely to
win each game, but do not know the exact probability. If the most likely number
of games played is 5, what does this imply about the probability that team A
wins each game?
2. Assume that individual income is uniformly distributed
between $0 and $100,000. For each of the following, plot the estimated
distribution using simulations.
(a) Suppose individuals select their spouse completely at
random. What is the distribution of total household income in this case.
(Mathematically, what is the distribution of X + Y where X, and Y are
independent ∼ Uniform(0, 100000)?)
(b) Suppose individuals are more likely to form pairs with
others of similar income. Describe and implement at least two ways of modeling
this phenomenon. What is the impact to the distribution of household income?
3. A trader is trying to decide whether stock A or stock B
will perform better next quarter. He seeks the opinions of two advisors. He
knows that each advisor will choose the better stock with probability p,
independent of the other. His strategy is as follows: if the two advisors
agree, buy the stock on which they agree. If they disagree, toss a fair coin to
decide which stock to buy. Evaluate his strategy.
4. Gamblers Ruin: A trader has $100 to invest. Each day he
decides on an amount to risk. With probability p he doubles whatever amount was
put at risk, and with probability q = 1 − p, he loses the investment. This
continues until he has lost everything or has $200. Experiment with different
risk amounts and values of p and recommend a strategy.
