We are asked to determine the value of $F(14) = P(X \leq 14)$, where $F(x)$ is the cumulative distribution function (CDF) of the random variable $X$ associated with the given probability density function (PDF) $f(x)$.

The CDF is computed by integrating the PDF from the lower bound (in this case, $-\infty$) up to $x$:

$F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) \, dt$

In this case, we know the density function $f(x)$ is non-zero only between $-31.7$ and $14$, and equals zero outside that range.

Thus, $F(14)$, which represents $P(X \leq 14)$, will include the entire probability distribution between $-31.7$ and $14$. Since $f(x)$ is a probability density function, the total area under the curve of $f(x)$ over its support must equal 1:

$F(14) = \int_{-31.7}^{14} f(x) \, dx = 1$

Hence, $F(14) = 1$.

Conclusion:

$F(14) = P(X \leq 14) = 1$

Would you like further details, or do you have any questions?

Here are 5 related questions:

1. How is the cumulative distribution function related to the probability density function?
2. What are the properties of a probability density function?
3. How would you compute $F(x)$ for other values of $x$?
4. Can the cumulative distribution function $F(x)$ ever decrease?
5. What happens to $F(x)$ as $x$ approaches infinity?

Tip: When integrating a PDF over its entire support, the result must always be 1, representing the total probability.