Discrete cumulative distribution function

Definition

A discrete cumulative distribution function (CDF) is a mathematical function that provides the probability that a discrete random variable takes on a value less than or equal to a specified value. It is a key concept that helps summarize the probabilities of a discrete random variable, allowing for an understanding of how probabilities accumulate as the variable increases.

5 Must Know Facts For Your Next Test

1. The discrete CDF is non-decreasing, meaning it never decreases as the value increases; it can stay constant or increase.
2. At the smallest value of the discrete random variable, the CDF equals 0, and it approaches 1 as the value goes to infinity.
3. The CDF can be computed by summing up the probabilities from the probability mass function (PMF) for all values up to and including the specified value.
4. For a discrete random variable with finite outcomes, the CDF will have jump discontinuities at each possible value of the variable.
5. The range of a discrete CDF is between 0 and 1, inclusive, reflecting the total probability of all possible outcomes.

Review Questions

• How does the discrete cumulative distribution function relate to the probability mass function?
  • The discrete cumulative distribution function is derived from the probability mass function by summing the probabilities for all outcomes less than or equal to a given value. This means that the CDF provides a cumulative perspective on probabilities, allowing for insights into how likely it is for a random variable to be at or below a certain point. Essentially, while the PMF gives probabilities for individual outcomes, the CDF aggregates these probabilities to show their accumulation.
• Discuss the significance of jump discontinuities in the context of a discrete cumulative distribution function and what they imply about a random variable.
  • Jump discontinuities in a discrete cumulative distribution function indicate points where there are specific probabilities assigned to particular outcomes. Each jump corresponds to the probability associated with that outcome, meaning that at these points, the probability increases as you include additional outcomes. This characteristic reflects how discrete random variables behave—unlike continuous variables, which have no jumps and are represented smoothly, discrete variables have distinct steps in their CDF.
• Evaluate how understanding the properties of a discrete cumulative distribution function can aid in making decisions based on probabilistic models.
  • Understanding the properties of a discrete cumulative distribution function allows one to interpret and analyze probabilistic models more effectively. By knowing how probabilities accumulate and where significant jumps occur, decision-makers can assess risks and make informed choices based on potential outcomes. For instance, if a decision involves selecting options based on their associated probabilities, knowing the CDF enables evaluation of likelihoods for various scenarios, ultimately guiding strategic planning and risk management.

Related terms

Discrete Random Variable:

A type of random variable that can take on a finite or countably infinite number of distinct values.

Probability Mass Function:

A function that gives the probability that a discrete random variable is exactly equal to some value.

Quantiles:

Values that divide a probability distribution into intervals with equal probabilities, commonly used in the context of CDFs.