The problem in the image asks to find the probability distribution function (PDF) given the cumulative distribution function (CDF) $F(x)$. The CDF is:

$F(x) = -0.0312x^3 + 0.187x^2$

The probability distribution function $f(x)$ is the derivative of the cumulative distribution function $F(x)$. This means:

$f(x) = \frac{d}{dx} \left[ F(x) \right]$

Now, we can differentiate the given function:

$f(x) = \frac{d}{dx} \left[ -0.0312x^3 + 0.187x^2 \right]$

Let me calculate this derivative.The probability distribution function (PDF) $f(x)$ is:

$f(x) = x(0.374 - 0.0936x)$

This expression corresponds to the derivative of the cumulative distribution function $F(x)$.

Would you like further clarification or more details on how this was derived?

Here are 5 related questions to explore:

1. What is the physical meaning of the PDF in this context?
2. How do we interpret the CDF for different values of $x$?
3. How can we verify if this PDF satisfies the properties of a valid probability distribution?
4. How is the expected lifetime of the circuit element calculated using the PDF?
5. What is the difference between PDF and CDF in practical applications?

Tip: To ensure that a function is a valid probability distribution function, it must integrate to 1 over its domain.