Final answer:
A probability density function (pdf) for a continuous random variable X with a constant value describes a uniform distribution, characterized by a constant probability over an interval, total probability of one, and equal probabilities for intervals of equal length.
Step-by-step explanation:
The characteristics and implications of a probability density function (pdf) for a continuous random variable X with a constant C essentially describe how the probabilities for different outcomes of the random variable are distributed over its range of possible values. If the value of the pdf is a non-negative constant C across a range where X can take values, we are dealing with a uniform distribution.
For a uniform distribution, the probability density function (pdf) is characterized by the following properties:
- f(x) > 0 since probabilities are always non-negative.
- The total area under the density function f(x) is equal to 1 because the sum of probabilities over all possible outcomes must equal 1.
- Since f(x) = C within some interval [a, b], the probability P(c < x < d) for c and d within [a, b] is found by calculating the area under the curve between c and d, which is just the base (d - c) times the height C, due to the rectangle shape.
- The cumulative distribution function (cdf) relates to the pdf in such a way that it gives the probability that the random variable X is less than or equal to a certain value.
An important implication of a pdf such as this is that the probabilities for intervals of equal length anywhere within [a, b] are the same because the pdf is constant (C). Additionally, for any continuous random variable, we should note that P(x = c) = 0, thus we never compute the probability of a single value but rather an interval.