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Probability & Statistics/exponential distribution

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Question
Is the exponential distribution (i.e. as in https://en.wikipedia.org/wiki/Exponential_distribution ) a "continuous probability distribution"?

If answer "no", can someone explain why not and yet why, say, a continuous uniform distribution is a "continuous probability distribution"?

Reason for my question:

Up until recently, I had always thought the word "continuous" in the term "continuous probability distribution" was indicating that the "probability distribution" is FOR a "continuous random variable". But then someone in a science forum made me doubt that and insists that it isn't but its there to indicate that the "function is continuous", but I didn't get his meaning. And, contrary to what I always thought, he insists that the exponential distribution is NOT a "continuous probability distribution" because it is "not continuous". But, again, don't get what he means. Is he right? If so, explain.

Answer
First: It says right in the introduction to the Wikipedia article that this is a continuous distribution. So your answer is right there. You are correct. Further, you are correct entirely. Your definition is entirely correct:

I had always thought the word "continuous" in the term "continuous probability distribution" was indicating that the "probability distribution" is FOR a "continuous random variable"

That is a true, correct, and essentially complete characterization of what it means to be a continuous probability distribution.

Now, to elaborate.

A continuous probability distribution is one in which the associated random variable is continuous (i.e. not discrete).

See:
If a random variable is a continuous variable , its probability distribution is called a continuous probability distribution. [1]

A continuous distribution describes the probabilities of the possible values of a continuous random variable. A continuous random variable is a random variable with a set of possible values (known as the range) that is infinite and uncountable. [2]

A random variable X taking values in set S is said to have a continuous distribution if P(X=x)=0 for all x∈S.[3]

When you work with the normal distribution, you need to keep in mind that it's a continuous distribution, not a discrete one. A continuous distribution's probability function takes the form of a continuous curve, and its random variable takes on an uncountably infinite number of possible values. This means the set of possible values is written as an interval, such as negative infinity to positive infinity, zero to infinity, or an interval like [0, 10]... [4]

A continuous probability distribution is a probability distribution that has a cumulative distribution function that is continuous.[5]

All of the above definitions are satisfied by the exponential distribution except the one from "Probability for Dummies" (second to last). I have underline the incorrect part of their definition. Perhaps your friend is a dummy? Perhaps "Probability for Dummies" was written by dummies? It is possible they meant to state what the last definition (from Wikipedia) says, which is that the cumulative distribution function is continuous. (Or perhaps simply that the domain of the probability distribution function is continuous, i.e. the random variable is continuous.)

The function that describes the probability distribution function of the exponential distribution is not a continuous function. It is zero for values x<0 and λe^(-λx) for x≥0, causing a jump discontinuity in the function.

However, that is not relevant to the definition of "continuous distribution" . The random variable in question takes on values on the real number line, which means it is a continuous random variable. It does not matter if the function that defines the PDF is continuous, especially since the CDF (which is really more important in that sense) will still be continuous.

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Clyde Oliver

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I can answer all questions up to, and including, graduate level mathematics. I do not have expertise in statistics (I can answer questions about the mathematical foundations of statistics). I am very much proficient in probability. I am not inclined to answer questions that appear to be homework, nor questions that are not meaningful or advanced in any way.

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