Final answer:
The question deals with the cumulative distribution function (CDF) used in continuous probability distributions, which helps calculate probabilities as areas under the probability density function (PDF). The exponential distribution example illustrates a specific continuous distribution with a memoryless property, and the area under the entire PDF is always equal to 1.
Step-by-step explanation:
The student's question pertains to the concept of a cumulative distribution function (CDF) in probability theory. The CDF of a random variable X is denoted as P(X ≤ x) and represents the probability that X will take a value less than or equal to x. For continuous distributions, the CDF can be used to find the probability that X takes on a value greater than x by calculating P(X > x) = 1 − P(X ≤ x), which corresponds to the area to the right of x under the probability density function (PDF).
In questions related to drawing a continuous probability distribution or identifying the proportion of probability associated with certain intervals, it is crucial to understand that for a continuous PDF, the probability at a single point is always zero, hence P(x = a) is always 0. When dealing with continuous probability functions, probabilities equate to areas under the curve defined by the PDF, and the total area under the PDF over its domain is always equal to 1, reflecting the property that the sum of all probabilities over the entire sample space is 1.
To illustrate with an example of a discrete probability density function (discrete PDF), consider a random variable X that can take on the values 0, 1, 2, 3, 4, 5 with their corresponding probabilities summing up to 1. This is a finite, countable set, distinguishing it from continuous distributions.
An exponential distribution is a type of continuous distribution characterized by its memoryless property and is commonly expressed as X~ Exp(m), where m is the decay parameter and the PDF is defined by f(x) = me−mx for x ≥ 0. This distribution has a unique property where the probability of an event occurring after a certain time does not depend on how long has already passed, provided that the event has not happened yet