13.1.7. Solved Examples

Try these first

1. 1.

   What is a Probability Density Function?

   Hint

   Think about how probabilities are represented for continuous variables.

2. 2.

   In a uniform distribution, is the probability of every outcome the same?

   Hint

   Consider how a flat curve looks on a graph!

3. 3.

   What does a Probability Density Function represent?

   • Probabilities at discrete points
   • Probabilities over intervals
   • Only the mean value

   Hint

   Think about how continuous distributions work!

4. 4.

   True or False: The total area under a PDF equals 1.

   • True
   • False

   Hint

   Consider the normalization property of PDFs.

5. 5.

   Given a PDF f(x) = 3x^2 for 0 ≤ x ≤ 1, calculate E[X] and P(X < 0.5).

   Hint

   Use the integration formula carefully for both calculations.

6. 6.

   A random variable X follows a triangular distribution with PDF f(x) = 2x for 0 ≤ x ≤ 1. Find the probability P(0.3 ≤ X ≤ 0.7) and the expected value E[X].

   Hint

   Set your integration limits appropriately based on the PDF.