7.2.3.1. Properties

Try these first

1. 1.

   What does the non-negativity property of a PDF state?

   Hint

   Think about what probabilities can be.

2. 2.

   Write the formula that expresses normalization for a PDF.

   Hint

   Consider the area under the PDF curve.

3. 3.

   What does the non-negativity property of a PDF imply?

   • Probabilities can be negative
   • Probabilities must be zero or greater
   • Probabilities sum to one

   Hint

   Reflect on the fundamental nature of probabilities.

4. 4.

   True or False: The integral of a PDF over its entire range must equal one.

   • True
   • False

   Hint

   Consider what it means for a total probability.

5. 5.

   Given a PDF defined as $f(x) = 3(1-x)^2$ for 0 ≤ x ≤ 1, determine whether it is a valid PDF and calculate its mean.

   Hint

   First, ensure the PDF meets the properties.

6. 6.

   If the variance of a random variable is known to be 4 and the mean is 5, what can you infer about the dispersion of data points?

   Hint

   Consider how variance represents spread from the mean.