6.4. Examples and Applications

Try these first

1. 1.

   What value can a discrete random variable take?

   Hint

   Think of counting outcomes.

2. 2.

   What is the expected value of rolling a fair six-sided die?

   Hint

   Add up all possible outcomes and divide by total outcomes.

3. 3.

   What is the PMF for a discrete random variable?

   • A function assigning probabilities to outcomes
   • A measure of dispersion
   • An expected value calculation

   Hint

   Think about how probabilities are distributed for outcomes.

4. 4.

   TRUE or FALSE: Continuous random variables can take any value within an interval.

   • True
   • False

   Hint

   Refer to the nature of continuous data.

5. 5.

   A factory produces light bulbs with a defect rate of 10%. Define a discrete random variable X representing the number of defective bulbs in a sample of 10. Calculate its expected value and variance.

   Hint

   Use the binomial distribution.

6. 6.

   A random variable X has a PDF defined as f(x) = 3x² for 0 ≤ x ≤ 1. Find the expectation and variance of X.

   Hint

   Remember to compute both E(X) and E[X²] for variance!