Final answer:
Continuous probability functions model the probabilities for continuous random variables using a pdf where the total area under the curve is 1. The probability of a variable x taking a specific value is zero; instead, probabilities for intervals are calculated. For example, P(x > 15) in a distribution ranging from 0 to 15 is zero.
Step-by-step explanation:
Understanding Continuous Probability Functions:
Continuous probability functions are used to describe the likelihood of outcomes for continuous random variables. In a continuous probability distribution, values are represented on a continuum, such as all points on a line segment. These distributions utilize a probability density function (pdf) where the area under the curve between two points (a and b) on a graph represents the probability (P) that the random variable x falls within that range (P(a ≤ x ≤ b)). The integral of the pdf over the entire range of x (from −∞ to +∞) sums to 1, signifying that the probability of x taking on a value within the specified range is certain.
When dealing with continuous random variables, the probability of the variable assuming any one specific value is 0, that is P(x = c) = 0 for any particular value c. Instead, probabilities are calculated for ranges of values. For a uniform distribution, which is a type of continuous distribution, the pdf is constant over the range of x. This is depicted as a rectangle when graphed, and calculating the probability for a subrange a to b involves finding the area under the pdf between these two points.
An example of calculating probability for a continuous distribution could be finding the P(x > 15) where the random variable x has a defined range (0 ≤ x ≤ 15). Here, since 15 is the upper bound of x, P(x > 15) is 0 because x cannot take on values greater than 15 in this distribution.
The complete question is: The following density function describes a random variable X. f (x) = 1 - if 0<x<2 = < 2 A. Find the probability that X is greater than 1.