Final answer:
The Cumulative Distribution Function (CDF) sums probabilities in a discrete distribution up to a point or integrates a Probability Density Function (PDF) in a continuous distribution to get probabilities. The PDF for continuous variables is a curve with total area equal to one, while for discrete variables, it lists probabilities for each outcome. CDF evaluates probability as the area under the curve, differing from discrete distributions that give probabilities for distinct points.
Step-by-step explanation:
The Cumulative Distribution Function (CDF) is a concept used in both discrete and continuous probability distributions. For a discrete random variable, the CDF sums the probabilities of all outcomes up to and including a particular value. In contrast, with continuous random variables, the CDF is integrated to find probabilities. The Probability Density Function (PDF) for a continuous random variable describes the density of probabilities across a continuous range and is represented as a curve on a graph where the total area under the curve is one. To find the probability between two points, you would calculate the area under the PDF curve between those points. However, for a discrete random variable, a Probability Distribution Function (PDF) also exists, which lists all possible outcomes and the probability associated with each outcome, where the sum of probabilities equals one.
The CDF for a continuous random variable is represented by P(X ≤ x), which is the integral of its PDF, f(x), from minus infinity to x. The whole area under the resultant CDF curve equals one, corresponding to the probability of the entire sample space. When working with CDF, the probability is evaluated as an area under the curve for a given interval or point, which stands in contrast to a discrete random variable where probability is associated with individual points or outcomes.