7.2.4. Properties of PDF

Try these first

1. 1.

   Explain what non-negativity means in the context of PDFs.

   Hint

   Think about the concept of probability.

2. 2.

   What is normalization of a PDF?

   Hint

   Consider how probabilities should sum up.

3. 3.

   What does normalization ensure for a PDF?

   • Total probability equals zero
   • Total probability equals one
   • PDF values can be negative

   Hint

   Remember that probabilities must always have a sum.

4. 4.

   True or False: The mean of a random variable gives us its most common value.

   • True
   • False

   Hint

   Consider what average means versus frequency.

5. 5.

   If a PDF is given as f(x) = 1/2 for 0 ≤ x ≤ 2, calculate the mean and variance.

   Hint

   You may use integration techniques.

6. 6.

   Find the probability that a random variable X, described by the PDF f(x) = 3x² for 0 ≤ x ≤ 1, lies between 0.5 and 1.

   Hint

   Remember to set up your integral limits correctly.