Introduction
Consider the following probability density function of a continuous probability distribution. Say it represents the time one may take to travel from point A to B.
For simplicity, we are assuming a uniform distribution in the interval [1,5].
Essentially, it says that it will take somewhere between 1 and 5 minutes to go from A to B. Never more, never less.
The Probability Density Function
Thus, the probability density function (PDF) can be written as follows:
fT(t) = { 1 / (5−1) : 1 ≤ t ≤ 5
0 : otherwise }
The Key Question
My question is: What is the probability that one will take precisely three minutes P(T=3) to reach point B?
Possible answers:
- A) 1/4 (or 0.25)
- B) Area under the curve from t=[1,3]
- C) Area under the curve from t=[3,5]
- D) It cannot be determined
The Correct Answer
Decide on an answer before you read further.
Well, all of the above answers are wrong.
The correct answer, however, is: ZERO.
And I intentionally kept only wrong answers here so that you never forget something fundamentally important about continuous probability distributions.
Let’s dive in!
Basics of Continuous Probability Distributions
The probability density function of a continuous probability distribution may look as follows:
Conditions for a Probability Density Function
- It should be defined for all real numbers (can be zero for some values).
- PDF → fX(x); x ∈ R
- The area should be 1.
- ∫−∞∞ fX(x) dx = 1
- The function should be non-negative for all real values.
- fX(x) ≥ 0 ∀ x
Common Misconception About PDFs
Here, many folks often misinterpret that the probability density function represents the probability of obtaining a specific value.
For instance, by looking at the above probability density function, many incorrectly conclude that the probability of the random variable X being 2 is close to 0.27.
P(X=2) = 0.27
But contrary to this common belief, a probability density function:
- DOES NOT depict the probabilities of a specific value.
- Is not meant to depict a discrete random variable.
The Actual Purpose of a PDF
Instead, a probability density function:
- Depicts the rate at which probabilities accumulate around each point.
- Is only meant to depict a continuous random variable.
Now, there are infinitely possible values that a continuous random variable may take.
So the probability of obtaining a specific value is always zero (or infinitesimally small).
Thus, answering our original question, the probability that one will take three minutes to reach point B is ZERO.
Why Do We Use PDFs?
So what is the purpose of using a probability density function?
In statistics, a PDF is used to calculate the probability over an interval of values.
Thus, we can use it to answer questions such as:
- What is the probability that it will take between 3 to 4 minutes to reach point B from point A?
- What is the probability that it will take between 2 to 4 minutes to reach point B from point A?
- And so on…
And we do this using integrals.
The Formula for Probability in an Interval
More formally, the probability that a random variable X will take values in the interval [a,b] is:
P(a ≤ X ≤ b) = ∫ab fX(x) dx
Simply put, it’s the area under the curve from [a,b].
Verifying That a Specific Value Has Probability Zero
From the above probability estimation over an interval, we can also verify that the probability of obtaining a specific value is indeed zero.
By substituting b = a, we get:
P(a ≤ X ≤ a) = ∫aa fX(x) dx = 0
Key Takeaways
So remember…
In a continuous probability distribution:
- The probability density function does not depict the exact probability of obtaining a specific value.
- Estimating the probability for a precise value of the random variable makes no sense because it is infinitesimally small.
- Instead, we use the probability density function to calculate the probability over an interval of values.
Conclusion
Understanding the difference between discrete and continuous probability distributions is essential in statistics.
The concept that the probability of any specific point in a continuous distribution is zero may sound counterintuitive at first, but it is one of the most important foundations of probability theory.
The probability density function (PDF) is not about the likelihood of single points, but rather about how probabilities accumulate over ranges.
So the next time you encounter a probability problem involving continuous variables, remember:
- Focus on intervals, not points.
- Use integrals, not simple fractions.
- And most importantly, never confuse density with probability.
