Final answer:
The cumulative distribution function (CDF) describes the probability of a continuous random variable being less than or equal to a specific value, representing areas under the probability density function curve. For specific point probabilities, such as P(x = 7), the value is zero since we are dealing with intervals in continuous distributions. To calculate probabilities like P(4 < x < 5), differences in CDF values are used.
Step-by-step explanation:
The cumulative distribution function (CDF) is a key concept in statistics used to describe the probability that a random variable will take a value less than or equal to a certain point. Specifically, for a continuous random variable, the CDF P(X ≤ x) represents the area under the probability density function (pdf) to the left of x. The entire area under the curve of the pdf and above the x-axis totals to one, encapsulating the foundational axiom that the sum of probabilities of all possible outcomes is equal to one. When evaluating probabilities for continuous random variables, it is important to understand that we are considering intervals rather than specific point values since the probability of a continuous random variable taking on any exact value is zero.
For example, if we want to calculate P(X > 15) for a continuous probability distribution within the interval 0 ≤ x ≤ 15, we know that this probability is zero because x cannot be greater than 15 within this distribution's defined bounds. Similarly, P(x = 7) or P(x = 10) would also be zero within their respective distributions.
In practice, the CDF is used to calculate probabilities by finding the area under the curve between certain bounds. For example, to find P(4 < x < 5), one might calculate P(x < 5) and subtract P(x < 4), with these values representing areas to the left of x = 5 and x = 4 respectively, given by the CDF.