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Distributions Calculators

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distributions

Normal PDF

Evaluate the normal probability density function at a point.

Example answer0.1295

This is the worked example that ships with the calculator. Replace the values above with your own and press Calculate.

x
1.5
Mean (μ)
0
Std Dev (σ)
1
  1. Normal PDF formulaFormulaf(x) = (1 / (σ√(2π))) * exp(-0.5 * ((x - μ) / σ)²)

    x = 1.5000, μ = 0.0000, σ = 1.0000

  2. StandardizeSubstitutez = (1.5000 - 0.0000) / 1.0000

    z = 1.5000

  3. Compute PDFSubstitute(1 / (1.0000 * 2.5066)) * exp(-0.5 * 1.5000²)

    0.1295

Free reference

Distributions: the formulas

Probability distributions assign probabilities to outcomes. The binomial distribution handles counts of successes in repeated trials, and the normal distribution handles continuous bell-shaped data through z-scores.

  • Binomial PMF: P(X = k) = C(n,k)p^k(1-p)^(n-k) - Exact probability of k successes in n independent Bernoulli trials.
  • Z-Score: z = (x - mu) / sigma - Standardizes a value by measuring standard deviations from the mean.

See the full formula reference

Worked example: Binomial PMF

A fair coin is flipped 10 times. What is the probability of getting exactly 3 heads?

Inputs. n = 10 trials, k = 3 successes, p = 0.5.

  1. Binomial PMF formula. P(X = k) = C(n,k) * pᵏ * (1-p)ⁿ⁻ᵏ gives k = 3, n = 10, p = 0.5000
  2. Compute binomial coefficient. C(10, 3) gives 120.0000
  3. Compute probability. 120.0000 * 0.5000^3 * 0.5000^7 gives 0.1172

Answer: 0.1172

There are C(10, 3) = 120 ways to place the 3 heads, each with probability 0.5 to the power 10, giving 120 / 1024, which is about an 11.7% chance.

Frequently asked questions

What is the difference between a PMF and a CDF?

The PMF gives the probability of exactly one value, such as exactly 3 heads in the worked example. The CDF accumulates: it gives the probability of that value or less, so the binomial CDF at 3 is the sum of the PMF at 0, 1, 2 and 3.

When is the binomial distribution appropriate?

When you count successes across a fixed number of independent trials that each have the same probability of success and only two outcomes. Ten flips of a fair coin meet all four conditions, which is why the example uses n = 10 and p = 0.5.

What does a z-score represent?

A z-score measures how many standard deviations a value sits from the mean: z = (x - mu) / sigma. A z of 2 means two standard deviations above the mean, and z-scores let any normal distribution be read off the standard normal table.