7.5. Probability Calculation

Try these first

1. 1.

   What does PDF stand for?

   Hint

   Recall the acronym we discussed.

2. 2.

   What property ensures the area under a PDF equals one?

   Hint

   Think about the key requirements of a PDF.

3. 3.

   What property must a PDF satisfy?

   • It can be negative
   • The total area under the curve is 0
   • The total area under the curve is 1

   Hint

   Consider what makes a valid probability distribution.

4. 4.

   Is the mean of a random variable always zero?

   • True
   • False

   Hint

   Think about different types of distributions.

5. 5.

   Given a continuous random variable with PDF f(x) = 12x(1-x) for 0 ≤ x ≤ 1, find the probability that X falls between 0.25 and 0.75.

   Hint

   Refer to integration techniques for polynomials.

6. 6.

   Calculate the expected value (mean) of X when its PDF is given as f(x) = 6x(1-x) for 0 ≤ x ≤ 1.

   Hint

   Distributing and integrating will be key here.