How to find probability distribution

Answer

Finding a probability distribution for a dataset or random variable involves determining whether the data is discrete or continuous, examining its shape through visual tools, applying goodness-of-fit tests across candidate distributions, and estimating parameters using methods like maximum likelihood estimation.

What is the process for finding a probability distribution?

To find a probability distribution, follow these steps:

1. Determine the data type by classifying the variable as discrete (countable values such as number of events) or continuous (any real value such as heights or weights).
2. Examine the data shape using histograms and summary statistics like skewness, kurtosis, and bounds to identify patterns.
3. Visualize the data with Q-Q plots, probability plots, and histograms to assess symmetry, number of modes, tail behavior, and bounds.
4. Apply goodness-of-fit tests across candidate distributions, prioritizing those with high p-values and strong visual fit.
5. Estimate parameters using maximum likelihood estimation or method of moments for the best-fitting distribution.

How do you assess data before choosing a distribution?

Initial data assessment requires plotting raw data using histograms to reveal characteristics such as skewness, multimodality, and bounds. Right-skewed histograms appear in income data, while left-skewed patterns occur near upper limits like purity percentages.

Computing descriptive statistics helps identify distribution characteristics. The mean approximately equaling the median suggests symmetry, while significant differences indicate skewness. Kurtosis values reveal tail heaviness, with higher values suggesting heavier tails than a normal distribution.

For discrete data, patterns like counts suggest Poisson distributions, while categorical trial outcomes suggest binomial distributions. Continuous data requires examination of whether values are bounded (positive-only values suggest exponential or gamma distributions) or unbounded (normal distribution may apply).

What are the main types of probability distributions?

Discrete distributions

Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials. Coin flips and defect counts represent typical applications. The distribution requires binary outcomes with fixed trials n and success probability p.

Poisson distribution represents counts of rare events over a fixed interval, such as customer arrivals or calls to a service center. This distribution suits positive integers with mean equal to variance and no upper bound.

Geometric and negative binomial distributions count trials until first success or r successes. These distributions fit waiting times or overdispersed count data that exceeds Poisson variance assumptions.

Continuous distributions

Normal (Gaussian) distribution produces a symmetric bell-shaped curve for continuous data around a mean. Heights, measurement errors, and many natural phenomena follow this distribution. The distribution is defined by mean μ and variance σ², with values unbounded in both directions.

Exponential distribution models time between Poisson events, such as machine failures or customer interarrival times. The distribution produces right-skewed, positive values with the memoryless property governed by rate parameter λ.

Uniform distribution assigns equal probability across a fixed interval. Random draws and simulation inputs use this distribution, producing flat density between bounds a and b.

Specialized distributions

Lognormal distribution applies to right-skewed positive data where the logarithm transforms to normal. Incomes, particle sizes, and multiplicative processes follow this distribution.

Gamma and Weibull distributions handle flexible right-skewed positive continuous data. Gamma distribution models sums of exponentials (rainfall accumulation), while Weibull distribution handles reliability and lifetime analysis with shape flexibility.

Beta distribution models bounded data on the interval [0,1], such as proportions and rates. Shape parameters α and β produce forms ranging from uniform to U-shaped or J-shaped curves.

How do visual tools help find probability distributions?

Histograms

Histograms display frequency distributions to identify skewness, modality, bounds, and peaks. A symmetric bell shape suggests normal distribution, while right-skewed patterns suggest exponential, lognormal, or gamma distributions. Bimodal histograms may indicate mixture distributions requiring further investigation.

Overlaying fitted densities on histograms provides visual confirmation of fit quality. Adjusting bin width prevents misleading artifacts that could suggest incorrect distribution shapes.

Q-Q plots

A Q-Q plot (quantile-quantile plot) is a graphical method for comparing two probability distributions by plotting their quantiles against each other. Ordered data quantiles plotted against theoretical quantiles of a candidate distribution reveal fit quality.

A point (x, y) on the plot corresponds to one of the quantiles of the second distribution plotted against the same quantile of the first distribution. Straight-line alignment indicates good fit, with the "fat pencil test" serving as a practical visual assessment method. Deviations from the line highlight mismatches in tails or center.

Normal QQ plots that look like curves instead of straight lines usually mean sample data are skewed. Points curving off in the extremities while following a line in the middle indicate heavy-tailed distributions with more extreme values than expected.

Probability plots

Probability plots are specialized Q-Q variants that linearize specific distributions. Normal probability plots transform normally distributed data into straight-line patterns. Points hugging the reference line confirm fit, while deviations indicate departures from the assumed distribution.

These plots excel at detecting tail behavior where statistical tests may underperform, particularly with small sample sizes.

Integration with statistical tests

Visual tools precede statistical tests for narrowing candidate distributions. Large samples make statistical tests overly sensitive, producing low p-values despite visually acceptable fits. Confirming test results visually ensures meaningful conclusions. A high Anderson-Darling p-value combined with an aligned Q-Q plot supports distribution acceptance.

What goodness-of-fit tests evaluate probability distributions?

Chi-squared goodness-of-fit test

The chi-square goodness-of-fit test can be applied to any univariate distribution for which you can calculate the cumulative distribution function. This test applies to discrete or binned continuous data, computing the test statistic using observed and expected frequencies.

The chi-squared test statistic formula is:

${\chi }^2$

where O_i represents observed frequencies and E_i represents expected frequencies under the fitted distribution.

A p-value greater than 0.05 indicates no significant deviation from the hypothesized distribution (good fit). The chi-square goodness-of-fit test is applied to binned data, and the value of the chi-square test statistic depends on how the data is binned. The test requires expected counts of at least 5 per bin and has reduced sensitivity in distribution tails.

Kolmogorov-Smirnov test

The Kolmogorov-Smirnov test is a nonparametric test of the equality of continuous one-dimensional probability distributions. The test compares the empirical cumulative distribution function to the theoretical cumulative distribution function via maximum vertical distance.

The Kolmogorov-Smirnov test statistic formula is:

$D=$

where F_n(x) is the empirical distribution function and F(x) is the theoretical cumulative distribution function.

A p-value greater than 0.05 suggests good fit. In practice, the statistic requires a relatively large number of data points compared to other goodness of fit criteria such as the Anderson-Darling test statistic to properly reject the null hypothesis. The test has higher sensitivity at the center of the distribution and lower sensitivity at the tails.

Interpretation guidelines

No single test provides definitive conclusions. Multiple tests combined with visual tools (Q-Q plots, probability plots) produce reliable assessments. Large samples often yield low p-values despite visually acceptable fits due to statistical power. Prioritize Anderson-Darling for continuous data in distribution fitting software. Reject the hypothesized distribution only when p-values fall below 0.05 and domain knowledge supports the decision.

How are distribution parameters estimated?

Maximum likelihood estimation

Maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data, achieved by maximizing a likelihood function so that the observed data is most probable under the assumed statistical model.

The likelihood function for independent observations is:

$L(\theta$

The log-likelihood function simplifies computation:

$\ell (\theta$

MLE finds parameter values that maximize this function, typically by setting derivatives equal to zero and solving. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. MLE provides consistency, efficiency, and asymptotic normality under standard regularity conditions.

Method of moments

This approach matches theoretical moments (mean, variance, skewness) of the distribution with sample moments calculated from data. The method is computationally simpler than MLE but may produce estimates outside valid parameter ranges.

For a distribution with parameters θ₁ and θ₂, the method solves:

${\mu }_1({\theta }_{}$

${\mathrm{<br>}}_1,{\theta }_2)=\overline{x}\text{and}{\mu }_2({\theta }_1,{\theta }_2)=\frac{1}{n}\sum _{i=1}^n(x_i-\overline{x}{)}^2$

Least squares estimation

Least squares methods minimize differences between observed and theoretical cumulative distribution functions. Ordinary least squares (OLS) weights all points equally, while weighted least squares (WLS) prioritizes certain data points. These methods prove useful for small samples or specific distribution fitting scenarios.

Additional estimation methods

Maximum product of spacings maximizes the product of spacings between ordered data points' CDF values, performing well for small to moderate sample sizes.

Bayesian estimation defines prior distributions on parameters and uses Bayes' rule to derive posterior distributions. Parameter estimates emerge as posterior means or modes, incorporating prior knowledge and quantifying uncertainty.

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