Final answer:
The height of a continuous probability function at any specific value of x is not the probability; the probability is the area under the curve over an interval. For any specific value, the probability is actually 0 for a continuous random variable.
Step-by-step explanation:
For a continuous random variable x, the height of the function at any specific value of x does not correspond to a probability. This is because for continuous random variables, the probability P(x = c) for any specific value c is 0. Instead, the probability is determined by the area under the probability density function (pdf) over an interval of x. Thus, if you want to find the probability that x falls within a certain range, you would calculate the area under the curve between the two bounds of that range.
For example, if we want to calculate P(x > 15) for a continuous probability distribution where 0 ≤ x ≤ 15, we know that this probability is 0 because x cannot be greater than 15 in this distribution. Similarly, if we want to find P(x = 7) for a continuous distribution where 0 ≤ x ≤ 10, this is also 0 because the value of a continuous random variable at any single point has zero probability.
To visualize this, imagine a graph of a continuous probability function, such as a uniform distribution, which can be represented by a rectangle. The total area of the rectangle under the probability function is 1, corresponding to the total probability. To find the probability between two points, you would calculate the area under the curve between these points.