Cumulative Distribution Function

Definition

The cumulative distribution function (CDF) is a fundamental concept in probability theory and statistics that describes the probability of a random variable taking a value less than or equal to a given value. It is a function that provides the cumulative probability distribution of a random variable, allowing for the calculation of probabilities for various ranges of values.

5 Must Know Facts For Your Next Test

1. The cumulative distribution function (CDF) provides the probability that a random variable is less than or equal to a given value.
2. The CDF is a non-decreasing function, meaning that as the value of the random variable increases, the CDF either stays the same or increases.
3. The CDF is closely related to the probability distribution function (PDF), as the CDF is the integral of the PDF over the range of the random variable.
4. The CDF is used to calculate probabilities for various ranges of values, such as the probability that a random variable falls within a specific interval.
5. The CDF is a fundamental concept in various probability distributions, including the discrete distributions (e.g., Binomial, Geometric, Hypergeometric, Poisson) and continuous distributions (e.g., Uniform, Normal, Chi-Square).

Review Questions

• Explain how the cumulative distribution function (CDF) is related to the probability distribution function (PDF) for a discrete random variable.
  • For a discrete random variable, the cumulative distribution function (CDF) is the sum of the probabilities of the random variable taking on values less than or equal to a given value. The CDF is the integral of the probability distribution function (PDF) over the range of the random variable. In other words, the CDF is the accumulation of the probabilities described by the PDF, providing the probability that the random variable is less than or equal to a specific value.
• Describe how the cumulative distribution function (CDF) can be used to calculate probabilities for a continuous random variable, such as in the context of the Normal distribution.
  • For a continuous random variable, the cumulative distribution function (CDF) provides the probability that the random variable is less than or equal to a given value. In the case of the Normal distribution, the CDF is represented by the standard Normal distribution function, denoted as $\Phi(z)$, where $z$ is the standardized value of the random variable. By using tables or computational methods to evaluate $\Phi(z)$, we can calculate the probability that a continuous random variable, such as a Normal random variable, falls within a specific range of values.
• Analyze how the properties of the cumulative distribution function (CDF), such as being non-decreasing and ranging from 0 to 1, are important for its use in various probability distributions and statistical applications.
  • The key properties of the cumulative distribution function (CDF) are that it is a non-decreasing function and its values range from 0 to 1. These properties are essential for the CDF to be a valid probability distribution function. The non-decreasing nature of the CDF ensures that as the value of the random variable increases, the cumulative probability also increases or remains the same. The range of 0 to 1 corresponds to the fundamental interpretation of probability, where 0 represents an impossible event and 1 represents a certain event. These properties of the CDF allow it to be used consistently across various probability distributions, both discrete and continuous, and enable the calculation of probabilities for different ranges of the random variable, which is crucial in statistical applications and decision-making.

Related terms

Probability Distribution Function (PDF): The probability distribution function (PDF) is a function that describes the relative likelihood of a random variable taking on a given value. It is the derivative of the cumulative distribution function.

Random Variable: A random variable is a variable whose value is subject to variations due to chance, and it can take on different values with certain probabilities.

Probability: Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.