6.3. Continuous Random Variables

Try these first

1. 1.

   What is a continuous random variable?

   Hint

   Think of examples like temperature or time.

2. 2.

   Which function represents the probability of a continuous RV?

   Hint

   Remember, it's a curve that describes probabilities.

3. 3.

   What is the primary purpose of a probability density function?

   • To provide a list of probabilities
   • To describe densities over intervals
   • To show discrete outcomes

   Hint

   Think about what a PDF represents graphically.

4. 4.

   The cumulative distribution function ranges from:

   • -∞ to +∞
   • 0 to 1
   • 0 to ∞

   Hint

   Consider how probabilities are represented in total.

5. 5.

   Given the PDF f(x) = 6x(1-x) for 0 ≤ x ≤ 1, calculate the expectation value E(X) and the variance Var(X).

   Hint

   First find the integral for E(X), then use the result to find Var(X).

6. 6.

   If a continuous random variable has a CDF F(x) = x^3 for 0 ≤ x ≤ 1, find the PDF and verify it integrates to 1.

   Hint

   Differentiate the CDF to find the PDF, then integrate.