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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.

What is the probability distribution function of the lifetime of the circuit element given the cumulative distribution function (CDF) F(x) = -0.0312x^3 + 0.187x^2?

This problem involves calculating the probability distribution function (PDF) of a circuit element's lifetime given the cumulative distribution function (CDF) F(x) = -0.0312x^3 + 0.187x^2. The solution uses basic calculus concepts and statistical theorems.

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