8.2.2. Continuous Random Variables

Try these first

1. 1.

   Define CDF and how it is used for continuous random variables.

   Hint

   Think of how you define probabilities.

2. 2.

   What does it mean for a function to be non-decreasing?

   Hint

   Consider how this looks on a graph.

3. 3.

   What is the main purpose of a CDF?

   • To describe the probability of a variable being greater than a value
   • To calculate the mean of a random variable
   • To describe the probability of a variable being less than or equal to a value

   Hint

   Think about what the CDF computes.

4. 4.

   Is the CDF for continuous variables a jump function?

   • True
   • False

   Hint

   Consider the properties of continuous functions.

5. 5.

   Given a continuous random variable with the PDF defined as f(x) = 6x(1-x) for x in [0,1], derive the CDF F(x).

   Hint

   Use integration by parts or the power rule.

6. 6.

   Consider a random variable X with the CDF defined as F(x) = x^3 for x in [0,1]. What is its PDF?

   Hint

   Recall the relationship between CDF and PDF.