Calculator
Enter matching values. The first probability belongs to the first x value, and so on.
Example Data Table
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P(X = x) | 0.10 | 0.25 | 0.30 | 0.25 | 0.10 |
This example has total probability 1. The distribution is symmetric around x = 2.
Formula Used
Probability rule: Each probability must satisfy P(X = x) ≥ 0, and the total must satisfy ΣP(X = x) = 1.
Expected value: E[X] = ΣxP(X = x).
Second moment: E[X²] = Σx²P(X = x).
Variance: Var(X) = E[X²] - (E[X])².
Standard deviation: σ = √Var(X).
Cumulative distribution: F(x) = P(X ≤ x) = ΣP(X = t), for all t ≤ x.
Entropy: H(X) = -Σp log₂(p), using only probabilities greater than zero.
How to Use This Calculator
- Enter all possible x values in the first box.
- Enter matching probabilities in the second box.
- Choose whether to normalize rounded probabilities.
- Select a custom probability condition if needed.
- Press the calculate button.
- Review the result above the form.
- Use CSV for spreadsheet work.
- Use PDF for a printable report.
Discrete Probability Distribution Guide
What It Measures
A discrete probability distribution lists every possible value of a random variable. It also gives the probability for each value. The probabilities must be nonnegative. Their total should equal one. This calculator checks those rules before it creates results.
Main Statistics
The mean shows the long run center of the distribution. It is also called the expected value. Variance measures average squared distance from that center. Standard deviation puts that spread back into original units. These three values are often the first review points in a statistics problem.
Cumulative Probability
This tool also calculates cumulative probability. The CDF answers questions such as P(X ≤ 3). It is useful when values have a natural order. You can also test a custom condition. Select less than, greater than, equal to, or between. Then enter the needed cutoff values.
Advanced Measures
Higher moments add more detail. Skewness shows whether the distribution leans left or right. Kurtosis describes tail weight and peak behavior. Entropy measures uncertainty. A higher entropy value means the probability is spread across more outcomes.
Input Advice
Use clean numeric inputs for best results. Put x values in one box. Put matching probabilities in the second box. Each probability belongs to the x value in the same position. The calculator can normalize probabilities when the total is positive but not exactly one. This is helpful after rounding.
Output Review
The table gives each x value, probability, cumulative probability, and survival probability. It also shows a z score when the standard deviation is positive. The download buttons save the current analysis. Use CSV for spreadsheets. Use PDF for a quick report.
Best Uses
This calculator is designed for homework, teaching, quality checks, and applied data work. It supports simple distributions and larger custom tables. Always confirm that the random variable is discrete. Continuous data needs a different model. Discrete distributions appear in counts, scores, defects, arrivals, claims, and survey choices. They also appear in games of chance. A careful table prevents hidden mistakes. It makes every assumption visible. That is why probability rows should be reviewed before interpreting the final numbers. Small rounding errors are common. Large gaps need correction. Keep notes about data sources, rounding choices, and any normalization applied during each analysis.
FAQs
1. What is a discrete probability distribution?
It is a table that lists possible values of a random variable and the probability of each value. The probabilities must be nonnegative and must total one.
2. What does the expected value mean?
The expected value is the weighted average of all x values. It shows the long run center if the same random process is repeated many times.
3. Why must probabilities add to one?
The full distribution must cover all possible outcomes. A total of one means the table represents 100 percent of the probability space.
4. What does normalization do?
Normalization divides each probability by the total probability. It is useful when rounded values are close to one but not exactly equal to one.
5. What is the CDF?
The CDF is the cumulative distribution function. It gives the probability that X is less than or equal to a selected value.
6. What does variance show?
Variance shows how far values spread around the expected value. A larger variance means outcomes are more widely scattered.
7. Can I use decimal x values?
Yes. The calculator accepts integer and decimal x values. Each value still needs a matching probability in the same position.
8. When should I use this calculator?
Use it when your random variable has countable outcomes, such as scores, counts, categories coded by number, or finite probability tables.