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A cumulative distribution function (CDF) is essential for describing the behavior of random variables in probability and statistics. One of the main purposes of a CDF is to define the probability that a random variable takes on a value less than or equal to a specific point. To be considered a valid CDF, a function must satisfy three important properties:

• It starts at zero: For any CDF function, the value at negative infinity must be zero, i.e., \( F(-\infty) = 0 \).
• It ends at one: As the variable approaches positive infinity, the CDF should reach one, so \( F(\infty) = 1 \).
• It is non-decreasing: The values should never decrease as you move from left to right along the x-axis. This means that the function must be non-decreasing for all real numbers.

Not abiding by these rules disqualifies a function from being a valid CDF, as demonstrated by the example function \( P(t) = e^{-t^2} \) which fails to meet these criteria.

The end behavior of a function provides critical insight into whether or not a function qualifies as a cumulative distribution function. When assessing the end behavior, you analyze what happens to the function as the variable approaches both negative and positive infinity.

For a valid cumulative distribution function:

• As \( t \) approaches \( -\infty \), the function should tend towards zero.
• As \( t \) moves towards \( \infty \), the function must approach one.

The example function \( P(t) = e^{-t^2} \) approaches zero at both extremes \(-\infty\) and \( \infty\), which contradicts the required boundary of reaching one as \( t \) approaches \( \infty \). Thus, it does not satisfy the end behavior criteria needed for a CDF.

Monotonicity is a key characteristic of a cumulative distribution function, ensuring that the function does not decrease as the variable increases. A CDF must be a non-decreasing function, which mathematically means that the slope, or derivative of the function, should be zero or positive for all values of the variable.

In the case of \( P(t) = e^{-t^2} \), its derivative, \( P'(t) = -2t \cdot e^{-t^2} \), presents issues with monotonicity:

• For \( t > 0 \), the derivative is negative, indicating a decreasing function.
• For \( t < 0 \), the derivative is positive, showing an increasing trend.
• At \( t = 0 \), the derivative is zero, offering brief monotonicity.

Because the derivative changes sign based on the value of \( t \), \( P(t) \) is not a non-decreasing function from \(-\infty\) to \( \infty\), thus invalidating it as a CDF.

Probability functions, specifically CDFs, are tools that ensure every possible outcome of a random variable can be captured under the probability scale of 0 to 1. This scale relies on the CDF's properties: correctly displaying the probability all possible events accumulate to one, starting at zero probability and ending at full probability, while adhering to monotonicity.

When a function does not meet this structure, it cannot provide a valid measure of total probability over the full spectrum of a random variable. The function \( P(t) = e^{-t^2} \) fails because it does not end at a probability of one as the variable approaches infinity.

• It never encompasses all probabilities since it begins and ends at zero.
• It cannot be considered a comprehensive representation of cumulative probabilities across an interval.

For a function to be a probability function, especially a CDF, these foundational requirements must consistently apply.