The given probability density function (PDF) of the component's life (in months) is:

$f(x) = \frac{1}{2} e^{-x/2}, \quad \text{for} \ x \in [0, \infty)$

This is an exponential distribution with a rate parameter $\lambda = \frac{1}{2}$.

We are asked to find the probability that the component lasts between 13 and 21 months. Mathematically, this is given by:

$P(13 \leq X \leq 21) = \int_{13}^{21} \frac{1}{2} e^{-x/2} \, dx$

I will now calculate this integral to find the probability.The probability that a randomly selected component will last between 13 and 21 months is approximately 0.00148, or about 0.148%.

Would you like further details on the integration process or any related questions?

Here are 5 related questions:

1. What is the cumulative distribution function (CDF) for this exponential distribution?
2. How would you find the expected value (mean) of the component's life?
3. What is the variance of the life of the component in this distribution?
4. How would the probability change if the range was extended to 13 to 30 months?
5. How can you interpret the result of such a small probability?

Tip: The exponential distribution is commonly used to model the time until an event occurs, such as the failure of a component.