Probability & Statistics Ch.2 Section 1 Question 11

· Mohammad-Ali Bandzar

Solutions for “Probability and Statistics: The Science of Uncertainty” (Second Edition). These are solutions I have come up with; I offer no guarantee of accuracy.

Question

Suppose the sample space S is finite. Is it possible to define an unbounded random variable on S? Why or why not?

Solution

We can find the definition of unbounded from Example 2.1.10: if X(s)X(s) increases or decreases without bound as s→∞s \rightarrow \infty, X can be called an unbounded random variable.

If our sample space is finite, we cannot have an unbounded random variable on S. This is because we must have:

  • An upper bound defined by max⁡s∈SX(s)\max_{s\in S}X(s)
  • A lower bound defined by min⁡s∈SX(s)\min_{s\in S}X(s)

Both of these exist for any finite sample space.