Probability distribution: P(X ≤ x)?
Pick a distribution, type its parameters and a value x. The probability distribution calculator shows the probability up to x and above it, the probability or density at x, the mean and the standard deviation, and draws the curve.
- P(X ≤ x)
- 0.8413447461
For this normal distribution, P(X ≤ 1) = 0.8413447461.
- P(X > x)
- 0.1586552539
- Density f(x)
- 0.2419707245
- Mean
- 0
- Standard deviation
- 1
- Variance
- 1
- Distribution
- normal
P(X ≤ x): 0.8413447461. For this normal distribution, P(X ≤ 1) = 0.8413447461.
Where is P(X ≤ x)?
How to calculate
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.
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.
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).
- 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
Each example is checked against the calculator on every build.
- Distribution Normal, Mean (μ) 0, Standard deviation (σ) 1, Value (x) 1.96 gives P(X ≤ x) 0.975002, P(X > x) 0.024998, Density f(x) 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
- Distribution Binomial, Trials (n) 10, Probability of success (p) 0.5, Value (x) 5 gives P(X = x) 0.246094, P(X ≤ x) 0.623047, P(X > x) 0.376953, Mean (μ) 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
- Distribution Poisson, Mean count (λ) 4, Value (x) 2 gives P(X = x) 0.146525, P(X ≤ x) 0.238103, Standard deviation (σ) 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
- Distribution Exponential, Mean (β) 2, Value (x) 1 gives P(X ≤ x) 0.393469, P(X > x) 0.606531, Density f(x) 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
- Distribution Uniform, Lower bound (A) 0, Upper bound (B) 10, Value (x) 2.5 gives P(X ≤ x) 0.25, Density f(x) 0.1, Mean (μ) 5, Standard deviation (σ) 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
How it works
Normal (mean μ, standard deviation σ > 0): P(X ≤ x) = Φ((x − μ) ÷ σ), where Φ is the standard normal cumulative distribution; P(X > x) = Φ((μ − x) ÷ σ); density f(x) = e^(−(x − μ)²/(2σ²)) ÷ (σ√(2π)); mean μ; standard deviation σ.
Binomial (n trials from 1 to 1,000, success probability p from 0 to 1, x a whole number): P(X = x) = C(n, x) pˣ (1 − p)ⁿ⁻ˣ; P(X ≤ x) = the sum for j = 0 to x. With p = a/D in lowest terms (the typed decimal), each probability is a whole number over Dⁿ, summed exactly and rounded once. Below 0: P(X ≤ x) = 0; at x ≥ n: P(X ≤ x) = 1, P(X = n) = pⁿ. Mean np; standard deviation √(np(1 − p)).
Poisson (mean count λ > 0, x a whole number): P(X = x) = e^−λ λˣ ÷ x!; P(X ≤ x) = Σ e^−λ λʲ ÷ j! for j = 0 to x, computed as the regularized upper incomplete gamma Q(x + 1, λ), and P(X > x) as P(x + 1, λ). Below 0: P(X ≤ x) = 0. Mean λ; standard deviation √λ.
Exponential (mean β > 0, starting at 0): P(X ≤ x) = 1 − e^(−x/β) for x ≥ 0 (0 below); P(X > x) = e^(−x/β); f(x) = e^(−x/β) ÷ β for x ≥ 0 (0 below). Mean and standard deviation β.
Uniform (from A to B, B > A): P(X ≤ x) = (x − A) ÷ (B − A) between A and B, 0 below A and 1 above B; f(x) = 1 ÷ (B − A) from A to B (0 outside). Mean (A + B) ÷ 2; standard deviation (B − A) ÷ √12.
Every result shows to 10 significant figures, rounded half up. The variance is the standard deviation squared. A binomial or Poisson x that is not a whole number, and B ≤ A, have no answer. The chart shades the area up to x.
Assumptions
- Parameters from −10⁹ to 10⁹ where they may be negative (μ, A, B, x); σ and β from 10⁻⁹ to 10⁹; λ from 10⁻⁹ to 100,000.
- Discrete probabilities other than the binomial and every continuous one are computed in double precision.
Worked examples by hand
Standard normal at 1.96. Φ(1.96) = 0.9750021; P(X > 1.96) = 0.0249979; f(1.96) = e^(−1.9208) ÷ √(2π) = 0.0584409.
Binomial, 10 tosses, p = 0.5, x = 5. C(10, 5) = 252; 252 ÷ 1,024 = 0.24609375; P(X ≤ 5) = (1 + 10 + 45 + 120 + 210 + 252) ÷ 1,024 = 638 ÷ 1,024 = 0.623046875; mean 5.
Poisson, λ = 4, x = 2. P(X = 2) = e⁻⁴ × 16 ÷ 2 = 8e⁻⁴ = 0.1465251; P(X ≤ 2) = e⁻⁴ (1 + 4 + 8) = 13e⁻⁴ = 0.2381033; standard deviation √4 = 2.
Exponential, mean 2, x = 1. 1 − e^(−0.5) = 0.3934693; e^(−0.5) = 0.6065307; f(1) = 0.6065307 ÷ 2 = 0.3032653.
Uniform from 0 to 10, x = 2.5. 2.5 ÷ 10 = 0.25; f = 1 ÷ 10 = 0.1; mean 5; standard deviation 10 ÷ √12 = 2.8867513.
Other questions people ask
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.