The Probability Mass Function (PMF) Calculator is an interactive tool designed to help users calculate and visualize the PMF for discrete random variables. The PMF provides the probability of each possible outcome in a discrete probability distribution, making it a fundamental concept in statistics and probability theory.

Key Features:

1. Dynamic Input : Users can input their dataset as comma-separated values along with their corresponding probabilities, and the tool will compute and display the PMF.
2. Validation : Ensures that the probabilities sum to 1, adhering to the rules of probability distributions.
3. PDF Download : Export the generated PMF table and visualization as a PDF for sharing or reporting purposes.
4. Modern Design : Clean and stylish layout with vibrant colors and clear typography for better readability.
5. Self-Contained : The tool operates within its own container, ensuring it doesn’t interfere with the page header or footer.

Use Cases:

• Educational Purposes : Teachers and students can use this tool to learn and teach probability mass functions.
• Data Analysis : Analysts can quickly compute and visualize PMFs for discrete datasets.
• Reporting : Professionals can export PMFs to include in reports or presentations.

How It Works:

1. Introduction : A Probability Mass Function (PMF) assigns probabilities to each possible value of a discrete random variable. For example, if $X$ is a random variable, the PMF $P(X=x)$ gives the probability that $X$ takes on the value $x$.
2. Input Fields : Users enter their dataset as two comma-separated lists: one for the outcomes (values of $X$) and one for their corresponding probabilities.
3. Validation : The tool ensures that the probabilities sum to 1. If not, it prompts the user to correct the input.
4. Visualization : The tool generates a bar chart to visualize the PMF.
5. Export : Users can download the PMF table and visualization as a PDF for further use.