Probstat/week5 practice 1

Practice exercises

Variances

1. Consider a random variable Z that becomes 1 with probability 1/3, 0 with probability 1/3, and 1 with probability 1/3. Find E[Z], and Var(Z).

2. Let random variable X be the outcome after tossing a fair die (with 6 faces of values 1,2,3,...,6). Find E[X] and Var(X).

3. You have observed the distribution of the score for a probability class. You notice that, from an exercise, students get 5 points with probability 0.1, 4 points with probability 0.2, 3 points with probability 0.3, 2 points with probability 0.1, 1 point with probability 0.2, and 0 points with probability 0.1. Let random variable X be the points a student will get. Find E[X] and Var(X).

Binomial and geometric random variables

1. Your friend claims to have a fair coin. You would like to prove it so you take her coin and toss it for 20 times. You got 15 heads. Suppose that the coin is truely a fair coin, what is the probability that this happens. Now, given that result, do you believe that the coin is indeed a fair one?

2.

3. An airline knows that a passenger will miss the flight independently with probability 0.05. Given an airplane with the capacity of 300 passengers, the airline would like to allow over-booking, i.e., it will allow the booking of more than 300 passengers. However, if they allow too many extra passengers, it might be the case that more than 300 passengers show up and they run into problems. What should be the number of extra passengers that the airline should allow to make sure that with probability at least 0.95, no more than 300 passengers will show up? (Notes: in this problem, to figure out the right value, it might be useful to write a computer program.)

Theoretical exercises

1. For a random variable X with variance Var(X). For a constant a, prove that $Var(aX)=a^{2}Var(X)$.

2. For a non-negative discrete random variable $X$ that takes on values 0,1,2,..., show that $\mathrm {E} [X]=\sum _{i=0}^{\infty }P\{X\geq i\}$.