Cumulative Distribution Function (CDF)

Definition

The Cumulative Distribution Function (CDF) is a fundamental concept in probability and statistics that describes the probability that a random variable takes a value less than or equal to a given value. It is a crucial tool for understanding and analyzing the behavior of random variables, particularly in the context of the Exponential Distribution.

5 Must Know Facts For Your Next Test

1. The CDF of a random variable $X$ is denoted as $F(x)$ and represents the probability that $X$ is less than or equal to a given value $x$.
2. For a continuous random variable, the CDF is the integral of the Probability Density Function (PDF) from negative infinity to the given value $x$.
3. The CDF is a non-decreasing function, meaning that as the value of $x$ increases, the value of $F(x)$ also increases or remains the same.
4. The CDF of an Exponential random variable with rate parameter $\lambda$ is given by $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$.
5. The CDF is useful for calculating probabilities, percentiles, and other statistical measures related to the distribution of a random variable.

Review Questions

• Explain the relationship between the Probability Density Function (PDF) and the Cumulative Distribution Function (CDF) for a continuous random variable.
  • For a continuous random variable $X$, the Cumulative Distribution Function (CDF) $F(x)$ is the integral of the Probability Density Function (PDF) $f(x)$ from negative infinity to the given value $x$. Mathematically, this can be expressed as $F(x) = \int_{-\infty}^{x} f(t) dt$. The CDF represents the probability that the random variable $X$ takes a value less than or equal to $x$, while the PDF describes the relative likelihood of $X$ taking on a specific value. The CDF is a non-decreasing function that provides a comprehensive picture of the distribution of the random variable.
• Describe the properties of the CDF and how they are useful in the context of the Exponential Distribution.
  • The Cumulative Distribution Function (CDF) has several important properties that make it a valuable tool in the study of the Exponential Distribution. First, the CDF is a non-decreasing function, meaning that as the value of the random variable $x$ increases, the value of the CDF $F(x)$ also increases or remains the same. This reflects the fact that the probability of a random variable being less than or equal to a given value cannot decrease as that value increases. Second, the CDF of an Exponential random variable with rate parameter $\lambda$ is given by $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$. This closed-form expression allows for easy calculation of probabilities and other statistical measures related to the Exponential Distribution. Finally, the CDF is useful for generating random variables from the Exponential Distribution through the Inverse CDF method, which involves finding the value of the random variable corresponding to a given probability.
• Explain how the Cumulative Distribution Function (CDF) can be used to calculate probabilities and make inferences about the Exponential Distribution.
  • The Cumulative Distribution Function (CDF) is a powerful tool for making inferences and calculating probabilities related to the Exponential Distribution. Since the CDF of an Exponential random variable $X$ with rate parameter $\lambda$ is given by $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$, we can use this expression to calculate the probability that $X$ is less than or equal to a given value $x$. For example, to find the probability that the time between events in a Poisson process (modeled by the Exponential Distribution) is less than or equal to 5 units, we can calculate $F(5) = 1 - e^{-\lambda \cdot 5}$. Additionally, the CDF can be used to find percentiles, which are useful for making inferences about the distribution of the random variable. By inverting the CDF, we can determine the value of the random variable corresponding to a given probability, allowing us to understand the behavior of the Exponential Distribution in greater detail.

Related terms

Probability Density Function (PDF): The Probability Density Function (PDF) is a function that describes the relative likelihood of a random variable taking on a given value. The CDF is derived from the PDF by integrating the PDF over the range of values up to the given value.

Exponential Distribution: The Exponential Distribution is a continuous probability distribution that models the time between events in a Poisson process, such as the arrival of customers in a queue or the occurrence of radioactive decays. The CDF is a key component in understanding and working with the Exponential Distribution.

Inverse CDF: The Inverse CDF, also known as the Quantile Function, is the function that gives the value of the random variable for a given probability. It is the inverse of the CDF and is useful for generating random variables from a given distribution.