91Ó°ÊÓ

The cumulative distribution function (CDF) is a fundamental concept in probability and statistics. It represents the probability that a continuous random variable takes a value less than or equal to a given point. The CDF is a non-decreasing function and ranges between 0 and 1. For a given probability density function (PDF) like \( f(x) \), the CDF \( F(x) \) is calculated by integrating the PDF from the lower bound of the variable up to \( x \).
For the exercise above, the PDF is \( f(x) = 3e^{-3x} \) for \( x > 0 \) and 0 for \( x \leq 0 \). Therefore, the CDF can be expressed as:
\[ F(x) = \int_{0}^{x} 3e^{-3t} \, dt = 1 - e^{-3x} \text{ for } x > 0, \] and \( F(x) = 0 \) for \( x \leq 0 \).

• The function \( F(x) = 1 - e^{-3x} \) shows that as \( x \) increases, \( e^{-3x} \) decreases, making \( F(x) \) approach 1.
• This signifies that as we move further along the \( x \)-axis, the probability of observing a value less than or equal to \( x \) increases.

The importance of the CDF lies in its ability to help us understand the distribution of data over a range.

Integration is a key mathematical process used to calculate areas under curves, which in the context of probability, relate to finding probabilities.
For the probability density function \( f(x) = 3e^{-3x} \), we need to integrate from 0 to infinity to ensure the total probability equals 1. This integration process confirms whether a function can serve as a valid probability density function.
In the step-by-step solution, integration was performed as follows:

• We defined \( u = -3x \) and thus \( du = -3 \, dx \), making \( dx = -\frac{du}{3} \).
• Changing variables allowed us to transform the integral \( \int 3e^{-3x} \, dx \) into \( \int -e^u \, du \).

This transformation is crucial because it simplifies the evaluation. Upon solving, we find that:
\[ \int_{0}^{\infty} 3e^{-3x} \, dx = 1 \]This outcome verifies that the function can represent a continuous random variable.

A continuous random variable is one that can take on an infinite number of possible values within a given range.
Unlike discrete random variables, which are countable, continuous variables are represented by probability density functions. These functions do not give probabilities for specific outcomes; instead, they provide a probability density, which must be integrated over a range to find actual probabilities.
In this exercise, the function \( f(x) = 3e^{-3x} \) describes a continuous random variable for \( x > 0 \).
Key points about continuous random variables include:

• The value of the probability density function at a single point is zero because there are infinitely many potential individual outcomes for a continuous random variable.
• To find the probability of the variable falling within a specified interval, we calculate the area under the curve of that interval using integration.

This is why confirming the total area under the PDF equals 1 is essential; it ensures that our conceptual understanding aligns with the mathematical model.This understanding is critical in fields such as statistics, physics, and engineering, where continuous distributions are commonplace.