# Probability distribution: P(X ≤ x)?

Computes P(X ≤ x), P(X > x), P(X = x) or the density, the mean and the standard deviation for a normal, binomial, Poisson, exponential or uniform distribution, with its curve.

- Page: https://www.acalculator.org/statistics/probability-distribution-calculator
- JSON spec: https://www.acalculator.org/statistics/probability-distribution-calculator.json
- Version: 1e24e4c88e67

## Default answer

Example with the default inputs (Distribution Normal, Mean (μ) 0, Standard deviation (σ) 1, Value (x) 1): For this normal distribution, P(X ≤ 1) = 0.8413447461.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| dist | Distribution | The probability distribution of X. |
| mu | Mean (μ) | The mean of the normal distribution. |
| sigma | Standard deviation (σ) | The spread of the normal distribution; at least 0.000000001. |
| n | Trials (n) | How many independent trials, from 1 to 1,000. |
| p | Probability of success (p) | The chance of a success on each trial, from 0 to 1. |
| lambda | Mean count (λ) | The average number of events in the interval; at least 0.000000001. |
| beta | Mean (β) | The mean (scale) of the exponential distribution, 1 ÷ rate; at least 0.000000001. |
| a | Lower bound (A) | The smallest value the uniform distribution takes. |
| b | Upper bound (B) | The largest value; more than A. |
| x | Value (x) | The value to find probabilities at. A whole number for the binomial and Poisson counts. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| atMost | P(X ≤ x) | The probability that X is at most x: the cumulative distribution function at x. |
| above | P(X > x) | The probability that X is more than x: 1 − P(X ≤ x). |
| exactly | P(X = x) | For a count (binomial, Poisson): the probability of exactly x. |
| pdf | Density f(x) | For a continuous distribution: the height of the density curve at x. |
| mean | Mean | The expected value of X. |
| sd | Standard deviation | The spread of X. |
| variance | Variance | The standard deviation squared. |
| name | Distribution | The distribution used. |

## Method

Normal: Φ((x − μ) ÷ σ); binomial: Σ C(n, j) pʲ (1 − p)ⁿ⁻ʲ for j ≤ x; Poisson: Σ e^−λ λʲ ÷ j! for j ≤ x; exponential: 1 − e^(−x/β); uniform: (x − A) ÷ (B − A).

## Assumptions

- Binomial and Poisson x are whole numbers (counts). Below 0 every count has P(X ≤ x) = 0.
- The exponential distribution starts at 0 (location 0) with mean β, as NIST’s standard form.

## Worked examples

1. dist = normal, mu = 0, sigma = 1, x = 1.96 gives atMost = 0.975002, above = 0.024998, pdf = 0.058441. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm.
2. dist = binomial, n = 10, p = 0.5, x = 5 gives exactly = 0.246094, atMost = 0.623047, above = 0.376953, mu = 5. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.18 Binomial Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm.
3. dist = poisson, lambda = 4, x = 2 gives exactly = 0.146525, atMost = 0.238103, sigma = 2. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm.
4. dist = exponential, beta = 2, x = 1 gives atMost = 0.393469, above = 0.606531, pdf = 0.303265. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.7 Exponential Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm.
5. dist = uniform, a = 0, b = 10, x = 2.5 gives atMost = 0.25, pdf = 0.1, mu = 5, sigma = 2.886751. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.2 Uniform Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3662.htm.

## FAQ

### What is a probability distribution?

It says how likely each value of a random quantity X is. A discrete distribution (binomial, Poisson) gives a probability to each count; a continuous one (normal, exponential, uniform) gives a density, and probabilities are areas under it.

### What is the difference between P(X ≤ x) and P(X = x)?

P(X ≤ x) is the cumulative probability, the chance that X is x or less. P(X = x) is the chance of exactly x, which only makes sense for counts. For 10 fair coin tosses, P(X = 5) = 0.2461 and P(X ≤ 5) = 0.6230.

### Which distribution should I use?

Binomial for the number of successes in n independent yes/no trials; Poisson for the number of events in an interval at a known average rate; normal for measurements that cluster around a mean; exponential for the waiting time between random events; uniform when every value in a range is equally likely.

### Why is P(X = x) zero for a continuous distribution?

A single point has no width, so it has no area under the density. The calculator shows the density f(x), the height of the curve, instead; probabilities come from P(X ≤ x).

### How do I find the probability between two values?

Subtract the cumulative probabilities: P(a < X ≤ b) = P(X ≤ b) − P(X ≤ a). Run the calculator at b and at a. For a count, P(X ≥ k) = 1 − P(X ≤ k − 1).

### What is P(X ≤ 1.96) for the standard normal?

About 0.975, so P(X > 1.96) is about 0.025. That is why 1.96 is the cut-off for a two-sided 95% interval.

## Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm (retrieved 2026-10-02)
- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.2 Uniform Distribution (f(x) = 1/(B − A); mean (A + B)/2; standard deviation √((B − A)²/12)). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3662.htm (retrieved 2026-10-02)
- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.7 Exponential Distribution (F(x) = 1 − e^(−x/β); mean and standard deviation β). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm (retrieved 2026-10-02)
- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.18 Binomial Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm (retrieved 2026-10-02)
- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.19 Poisson Distribution (p(x; λ) = e^−λ λˣ / x!; mean λ, standard deviation √λ). https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm (retrieved 2026-10-02)