Question

A lottery ticket costs 2 dollars and the probability of winning is 0.1. If you win a lottery you receive 10 dollars, if you lose you receive 0 dollars.

  1. Let \(X\) be a random variable that represents your money gain after playing one round of lottery, i.e. \(X=8\) if you win and \(X=-2\) if you lose. Find \(a\) and \(b\) such that \(X=a\cdot Y+b\) where \(Y\sim Bernoulli(p)\), i.e. \(Y\) is a Bernoulli random variable. What is the value of \(p\)?
  1. Use the properties of expectation and variance to find \(E(X)\) and \(Var(X).\)
  1. If you play this lottery many-many times, do you think your average money gain will be positive or negative?
  1. You decided to test your luck and bought 5 lottery tickets. Let \(Z\) denote the number of winning tickets. What is the expectation and variance of \(Z\)? Hint: use the link between Binomial and Bernoulli random variables.
  1. What is the probability that at least one of these five tickets will win?
  1. Let \(W\) be the average money gain for your five tickets. Find the expectation and variance of \(W\). Hint: use \(X_1, \ldots X_5\) to represent the money gain of each ticket and find the formula that expresses \(W\) in terms of \(X_1,\ldots,X_5\)
  1. Find the chances that your average money gain is not negative, i.e. \(P(W\geq0)\)? Is it higher than 50%? Hint: first find the formula that expresses \(W\) in terms of \(Z\).
  1. Now you decided to buy 100 tickets. Let \(W\) be the average money gain for your 100 tickets. What is the expectation and variance of \(W\)?
  1. What is the approximate distribution of \(W\)?
  1. Use the answer in 9 to find the chances that your new average money gain is not negative, i.e. \(P(W\geq0)\)? Is it higher than 50%?
  1. Use the 68–95–99.7 rule to find the interval \([c,d]\) that contains 95% of \(W\) values, i.e. such that \(P(c\leq W\leq d)=0.95.\)
  1. Use the 68–95–99.7 rule to find 2.5-th percentile for \(W\). In other words we need to find the value \(t\) such that 2.5% of \(W\) values are less than \(t\), i.e. \(P(W\leq t)=0.025.\)
  1. Use standardization and the distribution table to find the 2.5-th percentile for \(W\).