Random Variables

Simulation of the dice experiment

Simulation of Buffon's coin experiment

Definition

Suppose that we have a random experiment with sample space S. A random variable X is a real-valued function defined on the sample space S of the experiment. More generally, a k-dimensional random vector X is a function from the sample space S into a subset T of R^k. Equivalently, X is a sequence of k random variables:

X = (X₁, X₂, ..., X_k)

Intuitively, you should think of a random variable as a measurement of interest in the context of the random experiment. Similarly, a random vector is a sequence of measurements. A random vector X is random in the sense that its value depends on the outcome of the experiment, which cannot be predicted with certainty before the experiment is run. Each time the experiment is run, an outcome s in S of the experiment occurs, and a given random vector X takes on the value X(s). In general, as you will see, the notation of probability suppresses references to the sample space.

1. Recall that the dice experiment consists of rolling a pair of (distinct) fair dice and recording the number spots showing on each die. Express each of the following random variables as a function defined on the sample space:

1. X = the score on the first die.
2. Y = the score on the second die.
3. Z = the sum of the two scores.
4. U = the minimum of the two scores.
5. V = the maximum of the two scores.

2. Run the simulation of the dice experiment 100 times. For each run, compute the value of each of the random variables in Exercise 1.

3. Recall that Buffon's coin experiment consists of tossing a coin with radius r < 1 randomly on a floor covered with square tiles of side length 1. The coordinates of the center of the coin are recorded relative to axes through the center of the square, parallel to the sides. Express each of the following random variables as a function defined on the sample space:

1. X = the x-coordinate of the coin center.
2. Y = the y-coordinate of the coin center.
3. Z = the distance from the center of the coin to the center of the square.
4. W = the distance from the center of the coin to the nearest square side.

4. Run the simulation of the Buffon's coin experiment 100 times with r = 0.2. For each run, compute the value of each of the random variables in Exercise 3.

Events

Suppose that X is a random vector for an experiment with sample space S, and that X takes values in a subset T of R^k. Any statement about the random vector is an event. More precisely, the events that can be defined in terms of X have the form

5. In the dice experiment, consider the variables defined in Exercise 1. Express each of the following events in the form {X in B} for a suitably chosen random vector X and subset B. Then express each event explicitly as a subset of the sample space.

1. {X < 3, Y > 4}
2. {Z = 7}
3. {U = V}

6. In Buffon's coin experiment, consider the random variables defined in Exercise 3. Express each of the following events in the form {X in B} for a suitably chosen random vector X and subset B. Then express each event explicitly as a subset of the sample space.

1. {X < Y}
2. {Z < 0.5}
3. {W > r}

In the following exercises, X is a random vector for an experiment with sample space S, and X takes values in a subset T of R^k. Suppose that B and C are subsets of T and B_i is a collection of subsets of T indexed by a countable set I.

7. Show that

8. Show that

9. Show that

10. Show that

11. Show that if B and C are disjoint, then so are

By Exercises 7-11, the mapping

from subsets of T to subsets of S preserves all of the set operations.

Probability distribution

Suppose again that X is a random vector for the experiment with sample space S and that X takes values in subset T of R^k.

12. Use the results of Exercises 7-11 to show that the mapping

on subsets of T defines a probability measure on T. This is called the probability distribution of X.

Thus, note that any random vector X for an experiment reproduces the basic structure of the probability space:

• A set of outcomes T (the possible values of X).
• A collection of events (the subsets of T).
• A probability measure on the events (the probability distribution of X).

13. In the dice experiment, consider the variables defined in Exercise 1. Find the probability of each of the following events (see also Exercise 5):

1. {X < 3, Y > 4}
2. {Z = 7}
3. {U = V}

14. Run the dice experiment 100 times. Compute the relative frequency of each of the events in Exercise 13.

15. In Buffon's coin experiment, consider the variables defined in Exercise 3. Find the probability of each of the following events (see also Exercise 5):

1. {X < Y}
2. {Z < 0.5}
3. {W > r}

16. Run Buffon's coin experiment 100 times. Compute the relative frequency of each of the events in Exercise 15.

Indicator variables

For any event A, there is a simple random variable called the indicator variable of A, whose value tells us whether or not A has occurred:

1_A(s) = 1 for s in A; 1_A(s) = 0 for s in A^c.

or more simply, in the language of the experiment,

1_A = 1 if A occurs; 1_A = 0 if A does not occur

17. Conversely, show that any random variable I that just takes the values 0 and 1 is the indicator variable of the event

18. In the dice experiment, let A denote the event that the sum of the scores is 7. Run the simulation 100 times, and for each run, compute the value of 1_A.

19. In Buffon's coin experiment, let A denote the event that the coin does not touch the sides of the square. Run the simulation 100 times with r = 0.2 and for each run, compute the value of 1_A.

20. Let A and B be events in a random experiment. Show that

21. Suppose that we have a collection of events A_i for an experiment, indexed by a countable set I. Show that

Basic and Derived Variables

Recall that the outcome of a random experiment is usually a sequence of a fixed number n of real measurements, so that the sample space S is a subset of Rⁿ. Thus, the outcome of the experiment itself can be thought of as a random vector. Let X denote the identity function on S:

X(s) = s for s in S

Then X is a random vector and the components of X are random variables that give the individual measurements. That is,

X = (X₁, X₂, ..., X_n) where X_i(s) = s_i for s in S.

The events that can be defined in terms of X are simply the events of the experiment; that is

If Y is another random vector for the experiment, taking values in a subset T of R^k, then Y is a function of X. That is, there is a function g from S into T such that

Y = g(X) so that Y(s) = g(X(s)) for s in S

We could refer to X as the basic outcome variable and Y as a derived variable.

22.In the dice experiment, consider the random variables defined in Exercise 1. By definition, the basic outcome vector is the vector of dice scores (X, Y). Express each of the following as a function of (X, Y).

1. Z
2. (U, V)

23.In Buffon's coin experiment, consider the random variables defined in Exercise 3. By definition, the basic outcome vector is the position of the coin center (X, Y). Express each of the following as a function of (X, Y).

1. Z
2. W

In many problems of elementary probability theory, the basic object of interest is a random vector X taking values in a subset T or R^k for some k. Whether X is the basic outcome vector or a derived vector is often irrelevant.

Probability Spaces