11.1.2.2. Central Moments

Try these first

1. 1.

   Define raw moment and give its formula.

   Hint

   Think about the expected value.

2. 2.

   What is the first central moment equal to?

   Hint

   Remember, it measures deviations from the mean.

3. 3.

   What does the first raw moment measure?

   • Skewness
   • Mean
   • Variance

   Hint

   Think about what E[X] represents.

4. 4.

   True or False: The second central moment equals variance.

   • True
   • False

   Hint

   Consider the essential definition of variance.

5. 5.

   A certain random variable X has E[X] = 4 and E[X^2] = 20. Calculate the variance using the relationship with raw moments.

   Hint

   Set up the variance formula carefully and remember the expected values.

6. 6.

   Discuss how the moment generating function can be applied in reliability engineering and provide an example.

   Hint

   Consider relationships between MGFs and lifetime data.