Continuous Probability Distribution Explained

What is a continuous probability distribution? It will be helpful to first define the terms continuous versus discrete, and then compare the two. According to Apple Dictionary, discrete means, “individually separate and distinct,” whereas continuous means, “forming an unbroken whole; without interruption.” A great way to visualize the difference is to look at a fretted versus a fretless bass:

discrete probability distribution

The distribution of playable notes on a fretted bass is discrete; you can play an F or an F-sharp, but you cannot play the tone in between these notes. Photo by Ehtan Prater.

continuous probability distribution

A fretless bass involves a continuous distribution of playable notes. As there are no frets demarcating half-steps,
a bassist can play an F, an F-sharp, and anything in between. Photo by Ethan Prater.

On a fretted bass, we have a discrete distribution of notes; a bassist can play an F or an F-sharp, but she cannot play the note in between these tones. On the other hand, a fretless bass features a continuous distribution of notes; not only can she play an F and an F-sharp, but she can play any tone in between. Let’s now see how this applies to probability.

Rolling two dice: A discrete probability distribution

Consider the roll of two standard, 6-sided dice. The outcome of the roll can be described by a discrete probability distribution, which looks as follows:

discrete probability distribution

A discrete probability distribution for the outcome of rolling two dice. Photo by Tim Stellmach.

In the probability distribution above, just like on the fretted bass, only certain values are possible. For example, when you roll two dice, you can roll a 4, or you can roll a 5, but you cannot roll a 4.5. The fact that this is a probability distribution refers to the fact that different outcomes have different likelihoods of occurring. For example (as any craps player knows), a 7 is the most common roll with two dice; we see this reflected in the distribution above, as 7 has the highest peak.

A giraffe’s neck: A continuous probability distribution

Now, consider the random variable of, say, the length of a giraffe’s neck. This variable can be described by a continuous probability distribution because the length of a giraffe’s neck could be 4 feet, or 5 feet, or 4.5 feet, or 4.2384 feet. The point is, the length of giraffe’s neck could, in theory, take any value between zero and infinity. Of course, just like with the dice above, the probability of different values will involve different likelihoods–and the likelihood that a giraffe has a neck taller than the Empire State Building will be close to nil.

continuous probability distribution

The length of giraffes’ necks can be described by a continuous probability distribution. Photo by Alexas_Photos.

Differences between discrete and continuous probability distributions

A major difference between discrete and continuous probability distributions is that for discrete distributions, we can find the probability for an exact value; for example, the probability of rolling a 7 is 1/6. However, for a continuous probability distribution, we must specify a range of values. That is to say, we cannot ask, “What is the probability that a giraffe has a neck of 5 feet?” On the other hand, we ca ask the question, “What is the probability that a giraffe has a neck between 4.5 and 5.5 feet?” The difference is in the type of distribution: discrete versus continuous.

For more info on continuous probability distributions, check out this page from Columbia’s PreMBA site. Otherwise, we’ll conclude by considering the normal distribution, which is the most familiar and commonly used continuous probability distribution.

The normal distribution: the most famous continuous probability distribution of all

Chances are, even if you haven’t heard the term “normal probability distribution,” you have probability heard the term “bell curve,” which refers to the same thing. It looks as follows:

continuous probability distribution

The normal distribution, a continuous probability distribution function. Photo by NIST.

The normal probability distribution is often an accurate assumption for a host of random variables out in the real world. For example, the heights and weights of any large adult population will be distributed normally (follow the bell curve). This means that the majority of the population has a height or weight close to the central mean (the peak of the distribution). Extreme heights and weights are rare, or have a low probability of being found (the shallow tails of the distribution). Of course, heights and weights are also other examples of continuous variables, since they can, in theory, take any value.

Are you still confused about discrete versus continuous probability distributions? If so, check out our statistics blog and videos!

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