8. Partial Differential Equations

Try these first

1. 1.

   What does the CDF represent mathematically?

   Hint

   Think about probability and how values are compared.

2. 2.

   What is the range of values that F(x) can take?

   Hint

   Remember it's related to probabilities.

3. 3.

   What does the CDF of a random variable X represent?

   • The probability that X equals a specific value.
   • The probability that X is less than or equal to a value x.
   • The average value of X.

   Hint

   Remember the definition of CDF.

4. 4.

   True or False: The CDF of a discrete random variable is a continuous function.

   • True
   • False

   Hint

   Think about how discrete variables behave.

5. 5.

   Given that a random variable X has the following probability density function: f(x) = 4x for 0 ≤ x ≤ 0.5, compute the CDF and evaluate F(0.25).

   Hint

   Integrate the PDF from the lower limit to 0.25.

6. 6.

   A manufacturer has two machines, A and B. The CDF for machine A's failure times is given. If the CDF for machine B's failure is a constant value of 0.3, discuss how this affects overall machine reliability.

   Hint

   Consider how the cumulative failure rate influences operational decisions.