Homework #4: Random Variables

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Problem 1

Recall Problem 1 of Homework #1, which defined the sum, difference, product, and quotient of the numbers produced by rolling a pair of dice. Which of these four are random variables? Explain why you chose the ones you did. (This is a trick question.)

Problem 2

Consider a different situation with dice. This time we have one die, but we roll it twice, and then add the two numbers. Carefully describe a model of this experiment which allows you to compute the probability of each value of the sum.

Compare this model to the model used in Problem 1 of Homework #1.

Problem 3

Now consider a new kind of problem. Suppose you have a die that you think is used to cheat. That is, inside of the die is a weight placed near one face of the die at the center of that face. That means that the face on the opposite side of the die is more likely to come up. However, you don't know which side is "loaded".

The image below shows an "unloaded" die on the left, and a die loaded near "5" (so that "2" is most likely to come up) on the right:


Describe a probabilistic model in which you can compute the probability of each side of the die coming up if rolled once based on the information above. Note: Even if you know the physics, you can't calculate a number without more information than is given here, so you will need one or more parameters in your model. (Use as many as is convenient, but make sure you define all of them.)

  1. What is your state space?

  2. What is the probability of each number?

    You probably need to use two different methods to assess the probability in your model.

  3. Now consider rolling the same (possibly "loaded") die twice. Think of this as two random variables with a common state space. What further assumption(s) do you need to make to compute the probability of each possible sequence of two rolls of one die?

  4. Compute the probability of each sequence of two rolls.

  5. Define the random variable X which gives the sum of two rolls carefully.

  6. Recall that the distribution of a random variable X:Ω → R is the function F:R → [0,1] defined by F(c) = P({ω∈Ω | X(ω) ≦ c}). Compute the distribution of the sum of two rolls.

  7. Suppose you are told that the die on the right in the image above is the actual case. What do you expect the probability of each number 1, ..., 6 that can appear in a single roll of the die is?

  8. For each value c = 1, ..., 6, compute P(X = c | E), where E is the event "the die is loaded on the 5 side".

Problem 4

  1. Compute the joint frequency distribution for each of the pairs from X, Y, and Z.
  2. Compute the joint frequency distribution of X, Y, and Z.
  3. There are obvious tables for presenting the pairwise distributions. Are there better and worse ways to present the joint distribution of all three variables?
  4. Why are dummy variables given the values 0 and 1, rather than 1 and 2?