91Ó°ÊÓ

A Cumulative Distribution Function (CDF) is a fundamental concept in probability theory. It's a function, often denoted as \( F(x) \), that gives the probability that a random variable \( X \) takes a value less than or equal to \( x \). In other words, it tells us how the probabilities accumulate as the values of \( X \) increase. This means for any given \( x \), \( F(x) \ = P(X \leq x) \).CDFs have a few key characteristics:

• They are non-decreasing, meaning as we move to the right on the \( x \)-axis, the value of \( F(x) \) does not decrease.
• They are right-continuous, meaning they do not exhibit sudden jumps when approached from the right.
• At the extreme ends, they have boundary behaviors: \( F(-\infty) = 0 \) and \( F(\\infty) = 1 \).

In the given exercise, the CDF is expressed as: \[F(x)=1-\exp \left(-\int_{0}^{x} g(u) d u\right)\]This represents how the continuous integral of the function \( g(u) \) accumulates probability up to \( x \). It allows us to understand how the function \( g(u) \) shapes the distribution of the random variable \( X \).

A non-decreasing function is a function that maintains or increases its value as its input increases. This property is crucial for cumulative distribution functions (CDFs). Since a CDF represents the accumulated probability of a random variable, if it were to decrease at any point, it would imply a negative probability, which is inconsistent with probability laws.In the context of the exercise, we have the integral of \( g(u) \) within the exponent of the CDF formula:\[F(x) = 1 - \exp \left(-\int_{0}^{x} g(u) du\right)\]To ensure this function is non-decreasing:

• The integrand \( g(u) \) must be non-negative (\( g(u) \geq 0 \)). If \( g(u) \) were negative in any part, the integral \( \int_{0}^{x} g(u) \, du \) could decrease for increasing \( x \), leading to a decrease in the CDF value \( F(x) \).

By assuring \( g(u) \geq 0 \), the cumulative nature of \( F(x) \) is maintained smoothly as \( x \) increases, adhering to the properties of a valid CDF.

Integral divergence, in this context, refers to the behavior of an improper integral which diverges instead of reaching a finite limit. For the Cumulative Distribution Function (CDF) to satisfy \( F(\infiy) = 1 \), it requires that the exponent becomes so large that \( \exp \left(-\int_{0}^{\finity} g(u) \, du\right) \) approaches zero.This is achieved when:

• The integral \( \int_{0}^{\finity} g(u) du \) diverges to infinity. This divergence ensures that the exponential term approaches zero, making \( 1 - \exp \left(-\int_{0}^{\finity} g(u) du\right) = 1 \).

Hence, the condition \( \int_{0}^{\finity} g(u) du = \inity \) is necessary to ensure that the entire probability distribution of \( X \) is covered. If this integral were finite, \( F(\finity) \) would be less than 1, implying that not all probability has been accounted for, which invalidates the function as a complete CDF. This principle is crucial to understanding how a distribution function must encompass the entire possible range of outcomes for a random variable.