Question

The random variable x describes the failure time (in years) of a certain hydraulic component. The function of the variable is:

f(x) = 32 / (x + 4)^3      x > 0

1) Determine if it is a function of density

2) Determine the cumulative probability function

3) Calculate the probability that the component's lifetime is between 2 and 6 years.

4) Calculate V (x) and E (x) for the life span of this type of component.

5) If you were to give a guarantee so that only 5% fail before the guarantee time. How long is the warranty time you can offer?

Answer 1

We would be looking at the first 4 questions here:

Q1) The sum of probabilities across the X range here is computed as:

As the sum of all probabilities here is 1.

Also, we know that:
4³ = 64

Therefore, (x + 4)³  > 64 for x > 0

Therefore, (x + 4)³ > 32 for x > 0

Therefore, 32 / (x + 4)³ < 1, which is the second property required to be a valid PDF. Therefore the given function is a density function here.

b) The cumulative function here is computed as:

This is the required cumulative probability function here as:

c) The required probability here is computed as:

P( 2 < X < 6)

Therefore 0.2844 is the required probability here.

d) The expected value is first computed here as:

We already know the second integral value as 1. Therefore, we have here:

Therefore E(X) = 4 here is the expected value of X.

Now to compute variance we first compute the second moment of X as:

Note that the first part of the integral here does not converge and therefore is not defined. therefore V(X) for thie PDF is not defined.