Probability Distributions

Pick a distribution, set its parameters, and query probabilities, percentiles (critical values) and moments, with its density/mass and cumulative curves.

Distribution

Mean (μ)

-1010

Standard deviation (σ)

0.110

Query

Cut value (x)

Query result

P(X ≤ x)

0.8413

P(X ≥ x)

0.1587

Distribution summary

Mean: 0
Variance: 1
Standard deviation: 1
Median: 0
Mode: 0
Skewness: 0
Kurtosis (excess): 0

The cut corresponds to a z-score z = (x − μ)/σ = 1 (this is what you look up in the “z table”).

Fundamentals & Explanation

What is a probability distribution?

It is the rule that spreads probability across the values a random variable can take. There are two main families:

Discrete (counts: 0, 1, 2…). Described by its mass function (PMF) $p(k)=P(X=k)$ . It measures the exact probability of an event.
Continuous (measures: any real). Described by its density (PDF) $f(x)$ . Here $P(X=x)=0$ ; what matters is the area under the curve.

Both share the cumulative function (CDF) $F(x)=P(X\le x)$ , rising from 0 to 1 by accumulating everything to the left. Almost every query is answered through the CDF.

Modeling Distributions

Normal (Gaussian) $\;N(\mu,\sigma)$ — The bell curve. Models errors and averages; foundation of the Central Limit Theorem.

f(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

Binomial $\;\text{Bin}(n,p)$ — Number of successes in $n$ independent trials with probability $p$ .

P(X=k)=\binom{n}{k}p^{k}(1-p)^{\,n-k}

Poisson $\;\text{Pois}(\lambda)$ — Count of rare events in a fixed interval, with mean rate $\lambda$ .

P(X=k)=\dfrac{\lambda^{k}e^{-\lambda}}{k!}

Exponential $\;\text{Exp}(\lambda)$ — Waiting time until the next Poisson event (same rate $\lambda$ ). It is memoryless: the chance of waiting longer does not depend on how long you've waited.

f(x)=\lambda e^{-\lambda x},\quad x\ge 0

Gamma $\;\text{Gamma}(\alpha,\beta)$ — Waiting time until the $\alpha$ -th event (sum of $\alpha$ exponentials); with shape $\alpha$ and rate $\beta$ . It is the conjugate prior for the rate of a Poisson/Exponential.

f(x)=\dfrac{\beta^{\alpha}}{\Gamma(\alpha)}\,x^{\alpha-1}e^{-\beta x}

Special cases: $\text{Exp}(\lambda)=\text{Gamma}(1,\lambda)$ and $\chi^2_\nu=\text{Gamma}(\nu/2,\,1/2)$ .

Beta $\;\text{Beta}(\alpha,\beta)$ — Distribution of a proportion on $[0,1]$ . It is the conjugate prior for a success probability, the Bayesian basis of A/B testing.

f(x)=\dfrac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)},\quad 0\le x\le 1

Sampling Distributions

These distributions are derived from the Normal and are used almost exclusively for Statistical Inference (hypothesis testing and confidence intervals). Instead of modeling natural phenomena, they model the behavior of statistics computed from samples.

Student's t ( $\nu$ df): Used for inference on means when the population variance is unknown and the sample is small. Has "heavier tails" than the Normal. As $\nu \to \infty$ , the t becomes identical to the Standard Normal.
Chi-square ( $\chi^2$ , $\nu$ df): Sum of $\nu$ squared standard normals. Only takes positive values. Used for goodness-of-fit, independence, and variance tests.
F of Fisher-Snedecor ( $d_1, d_2$ df): Ratio of two scaled $\chi^2$ variables. Primarily used in ANOVA (Analysis of Variance) and regression to compare variances.

How to interpret the calculator?

By value: Enter a cut and obtain the tail area $P(X\le x)$ or $P(X\ge x)$ . In statistics, this corresponds to calculating the p-value.
By interval: Two limits → $P(a\le X\le b)$ or the probability of landing outside. Useful for confidence bounds.
By probability (quantile): Enter an area (e.g. $\alpha=0.05$ ) and it returns the critical value on the horizontal axis. This is equivalent to looking up values in traditional printed tables.

Critical Values & Tables

Historically, t, $\chi^2$ and F values were looked up in book appendices. The calculator in "By probability" mode replaces these tables.

If you are running a hypothesis test at the 5% significance level for a right tail, simply select "Right Tail", input $0.05$ , and you'll get the exact critical value to reject $H_0$ .

Common z values (Normal)

Confidence level	$z^\*$ (two tails)
90%	1.645
95%	1.960
98%	2.326
99%	2.576

Distribution Moments

Dist.	Mean	Variance
Modeling
Normal	μ	σ²
Binomial	np	np(1−p)
Poisson	λ	λ
Exponential	1/λ	1/λ²
Gamma	α/β	α/β²
Beta	α/(α+β)	…
Sampling
Student's t	0	ν/(ν-2)
Chi-sq.	ν	2ν
F Fisher	d₂/(d₂-2)	...

* Some variances and means in t and F are only defined for large degrees of freedom. In the Poisson, mean and variance coincide.

Probability Distributions

Query

Query result

Distribution summary

Fundamentals & Explanation

What is a probability distribution?

Modeling Distributions

Sampling Distributions

How to interpret the calculator?

Critical Values & Tables

Common z values (Normal)

Distribution Moments

You may also like:

What is a probability distribution?

Modeling Distributions

Sampling Distributions

How to interpret the calculator?

Critical Values & Tables

Common z values (Normal)

Distribution Moments