Probstat/week5 practice 1

This is part of probstat.

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. (IPSES, ex 3.1a) A thumb drive produced by a certain company is defective with probability 0.01. What is the probability that a person buys 10 thumb drives and get 1 defective drive?

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.)

Geometric random variables

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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 $\mathrm {E} [X]=\sum _{i=0}^{\infty }P\{X\geq i\}$.

Programming exercises

The fullest bins

Perform the balls-and-bins experiment on various values of n = 10,50,100,200,500,1000,2000,5000,10000,20000. For each experiment, you should record the maximum load X.

For each value of n, perform 10 independent experiments and record the average maximum load, the minimum X (over these 10 experiments), and the minimum X (over these 10 experiments). Plot a graph. See how the values of X grow.