Stephen Tashi said:How do your course materials define a "bounded random variable"? The claim "if X is a bounded random variable, then E(X) exists" isn't true using the usual definition of a "bounded random variable".
Perhaps your problem assumes the domain of X is bounded as well as the range.
number0 said:The book specifically defined X as bounded as the following:
|X| < M < ∞ .
I do not know about the range though.
A bounded random variable is one that has a finite range and cannot take on values beyond a certain limit. This means that the values of the random variable are restricted to a specific interval or set of numbers.
The expected value of a random variable is a measure of its central tendency or average value. It is important to show that a bounded random variable has a defined expected value because it helps in understanding the behavior of the variable and making predictions based on its expected value.
If a random variable is bounded, it means that its values are limited to a specific range. This also means that the expected value of the variable will fall within this range. In other words, the expected value of a bounded random variable will always exist and be within the bounds of the variable's range.
To show that a random variable is bounded, one can provide evidence such as the variable's range and any restrictions on its values. This can be done through mathematical proofs or by demonstrating that the variable cannot take on values beyond a certain limit.
Proving that a bounded random variable has a defined expected value is necessary because it provides evidence of the variable's stability and predictability. It also ensures that the variable can be used in statistical analyses and calculations, as the expected value is a crucial component in many statistical formulas and methods.