Probability Density Function

Solution & Explanation

Can a Probability Density Function (PDF) Exceed a Value of 1?

Yes, a Probability Density Function (PDF) can indeed exceed a value of 1. This might seem counterintuitive at first because we often associate probabilities with values between 0 and 1. However, it's crucial to understand the role and interpretation of a PDF in the context of continuous random variables.

Understanding the PDF

1. Definition of PDF:

   • A PDF is a function that describes the likelihood of a continuous random variable taking on a particular value. It is defined for continuous distributions and provides a density rather than a probability.

2. Role of the PDF:

   • The PDF does not give the probability of the random variable assuming a specific value. Instead, it provides a measure of the relative likelihood of the random variable being near a particular value.

3. Area Under the Curve:

   • The critical aspect of a PDF is that the area under the curve over any interval represents the probability of the random variable falling within that interval.
   • The total area under the entire PDF curve over its range is always equal to 1, ensuring that the total probability across all possible outcomes is 100%.

Why PDF Values Can Exceed 1

1. Relative Likelihood:

   • The PDF indicates how concentrated the probability is over a small interval. If the interval is very narrow, the PDF can take on large values to ensure that the total area under the curve remains 1.

2. Example of Uniform Distribution:

   • Consider a uniform distribution over the interval $[0, 0.5]$. The PDF for this distribution is $f(x) = 2$ for $0 \leq x \leq 0.5$ and $f(x) = 0$ elsewhere.
   • Here, the PDF value of 2 ensures that the area under the curve over the interval $[0, 0.5]$ is 1: $\int_{0}^{0.5} 2 \, dx = 1$
   • The high value of the PDF reflects the concentration of probability over a small interval.

3. Interpretation:

   • A PDF value greater than 1 indicates a high density of probability over a small range of values. It does not imply that the probability of the random variable taking on a specific value is greater than 1.

Key Points to Remember

• The PDF can exceed 1, but this does not violate any probability principles.
• The probability of the random variable falling within a specific range is determined by the area under the PDF curve over that range, not the height of the curve.
• The total area under the PDF curve over the entire range of the random variable must always equal 1.

By understanding these concepts, we can correctly interpret and utilize PDFs in the analysis of continuous random variables, ensuring that we make valid inferences about the underlying probability distributions.