Question

Explain the difference between a probability mass function (PMF) and a probability density function (PDF).

Answer 1

Difference between a Probability Mass Function (PMF) and a Probability Density Function (PDF)

Probability Mass Function (PMF) and Probability Density Function (PDF) are both mathematical concepts used in probability theory to describe the probability distributions of random variables. However, they have some fundamental differences, as explained below:

1. Definition:

   • PMF: A PMF is used to describe the probability distribution of a discrete random variable. It assigns probabilities to each possible value of the random variable.
   • PDF: A PDF is used to describe the probability distribution of a continuous random variable. It provides the relative likelihood of the random variable taking on different values within a range.

2. Value Assignments:

   • PMF: A PMF assigns probabilities to individual values of a discrete random variable. It gives the exact probability of each outcome.
   • PDF: A PDF gives the relative likelihood (density) of a continuous random variable taking on a particular value. It does not provide the exact probability of any specific value but represents the probability density over an interval.

3. Probability Calculation:

   • PMF: For a discrete random variable, the probability of an event is calculated by summing the probabilities of the individual outcomes. The sum of all probabilities in a PMF is equal to 1.
   • PDF: For a continuous random variable, the probability of an event is calculated by integrating the PDF over the range of values corresponding to that event. The integral of the PDF over the entire range is equal to 1.

4. Graphical Representation:

   • PMF: A PMF is typically represented as a probability distribution table or a bar chart, where the values of the random variable are plotted on the x-axis, and their corresponding probabilities are plotted on the y-axis.
   • PDF: A PDF is typically represented as a smooth curve on a continuous scale, where the x-axis represents the values of the random variable, and the y-axis represents the probability density.

5. Support:

   • PMF: The support of a PMF consists of discrete values, and the probability is assigned only to those specific values.
   • PDF: The support of a PDF consists of an infinite range of values within a continuous interval, and the probability is assigned to different ranges of values rather than individual points.

6. Normalization:

   • PMF: A PMF is already normalized by design since the sum of all probabilities must be equal to 1.
   • PDF: A PDF may not be normalized by default. The integral of the PDF needs to be calculated to ensure that the total probability is equal to 1.

In summary, the main difference between a PMF and a PDF lies in the type of random variable they describe and the way they assign probabilities. PMFs are used for discrete random variables, assigning probabilities to individual values, while PDFs are used for continuous random variables, describing the relative likelihood of values within a range.