8.2. CDF for Discrete and Continuous Random Variables

Try these first

1. 1.

   Define what a CDF is.

   Hint

   Think about what the function describes.

2. 2.

   What is the range of values for a CDF?

   Hint

   What are the limits of probabilities?

3. 3.

   What does the CDF for a discrete random variable represent?

   • P(X < x)
   • P(X ≤ x)
   • P(X > x)

   Hint

   Remember what the CDF is defined as.

4. 4.

   True or False: The CDF can decrease as x increases.

   • True
   • False

   Hint

   Think about the probability structure.

5. 5.

   Prove that the CDF of a discrete random variable must approach 1 as x approaches infinity.

   Hint

   Consider how cumulative probabilities work with respect to the definition of total probability.

6. 6.

   For a continuous random variable defined by f(x) = 4x^(3) on [0,1], find the CDF at x = 0.75 and discuss the behavior near 1.

   Hint

   Set up your integral for the PDF correctly, and consider limits at the endpoint.