Suppose the density function for a random variable X is given by f(x) .

It is known that the density function f(x) > 0 between -31.7 and 14 . Outside of that range, f(x) equals zero.

( Translation, the density curve is above the x-axis between the numbers  -31.7 and 14. Otherwise, the density curve is the x-axis )

Determine F( 14 ) = P( X<= 14 )

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.

Suppose the density function for a random variable X is given by f(x). It is known that the density function f(x) > 0 between -31.7 and 14. Outside of that range, f(x) equals zero. Determine F(14) = P(X <= 14).

This problem involves determining the cumulative distribution function (CDF) for a random variable X with a known probability density function (PDF) f(x) that is non-zero between -31.7 and 14. Learn how to calculate F(14) using integration and the properties of probability distributions. Suitable for high school and college-level probability students.

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