Continuous distribution
Continuous distribution (also called continuous probability distribution) describes a probability distribution where the random variable can assume any value within a given interval. Unlike discrete distributions that assign probabilities to specific values, continuous distributions spread probability across infinite possible outcomes. Height, weight, time, and temperature are examples of measurements that follow continuous distributions[1].
The probability that a continuous random variable equals any exact value is technically zero. Probabilities are instead calculated for ranges of values. The probability that a randomly selected adult male weighs between 160 and 170 pounds makes sense; the probability that he weighs exactly 165.000000 pounds equals zero.
Historical development
The normal distribution, the most famous continuous distribution, has a rich mathematical history. Abraham de Moivre first introduced it in 1733 while studying approximations to the binomial distribution. His work appeared in The Doctrine of Chances but received limited attention initially.
Pierre-Simon Laplace investigated the mathematical properties of this distribution in 1774 and again in 1783, applying it to measurement errors[2]. Laplace developed much of the underlying theory. Carl Friedrich Gauss employed the same distribution in 1809 while analyzing astronomical observations. Through a historical quirk, the distribution became primarily associated with Gauss rather than de Moivre or Laplace.
Gauss derived the probability density function from simple assumptions about measurement error. He reasoned that small errors should be more likely than large ones, that positive and negative errors of the same magnitude should be equally probable, and that the mean of multiple measurements should be the most likely value. These seemingly modest assumptions led directly to the bell curve formula.
Laplace proved the central limit theorem, demonstrating that sums of random variables converge to the normal distribution regardless of the underlying distributions. This result, combined with Gauss's work on least squares, established the normal distribution's central role in statistics.
Mathematical foundations
A continuous probability distribution is defined by its probability density function (PDF). The PDF, typically denoted f(x), describes how probability is distributed across possible values. Unlike discrete probability mass functions, the PDF does not give probabilities directly[3].
Key properties of continuous distributions include:
The area under the PDF curve equals 1. This reflects the certainty that the random variable will take some value within its domain.
Probabilities are calculated by integrating the PDF over intervals. The probability that X falls between a and b equals the integral of f(x) from a to b.
The cumulative distribution function (CDF), denoted F(x), gives the probability that the random variable is less than or equal to x. The CDF equals the integral of the PDF from negative infinity to x.
Expected value and variance describe the center and spread of the distribution. These are calculated through integration of the PDF weighted by appropriate terms.
The normal distribution
The normal distribution (Gaussian distribution) dominates statistical practice. Its PDF follows the famous bell curve formula:
f(x) = (1 / sqrt(2 pi sigma^2)) * e^(-(x-mu)^2 / (2*sigma^2))
Here mu represents the mean and sigma the standard deviation. The distribution is symmetric around the mean, with tails extending infinitely in both directions.
Characteristics of the normal distribution:
- Approximately 68 percent of values fall within one standard deviation of the mean
- About 95 percent fall within two standard deviations
- Nearly 99.7 percent fall within three standard deviations (the empirical rule)
Many natural phenomena approximate normal distributions. Heights and weights in large populations, measurement errors in scientific experiments, and standardized test scores often follow bell curves. The central limit theorem explains why: quantities that result from summing many small independent effects tend toward normality.
The standard normal distribution has mean 0 and standard deviation 1. Any normal distribution can be transformed to standard form by subtracting the mean and dividing by the standard deviation. This standardization enables use of common statistical tables.
Other common continuous distributions
Several other continuous distributions appear frequently in statistical applications:
Uniform distribution - All values within a defined interval are equally likely. If X is uniform on [a, b], its PDF equals 1/(b-a) throughout that range. Random number generators typically produce uniform distributions, which are then transformed to generate other distributions.
Exponential distribution - Models waiting times between random events. The time between customer arrivals at a store or the lifespan of electronic components often follows exponential distributions. The distribution is memoryless: the probability of waiting another hour is the same whether you have waited five minutes or five hours.
Student's t-distribution - Similar to the normal distribution but with heavier tails. William Sealy Gosset developed this distribution in 1908 while working for Guinness Brewery, publishing under the pseudonym Student. The t-distribution is used when estimating population means from small samples with unknown variance.
Chi-squared distribution - Arises when summing squares of independent standard normal variables. This distribution is essential for hypothesis testing about variances and for goodness-of-fit tests.
F-distribution - Defined as the ratio of two independent chi-squared variables divided by their degrees of freedom. Analysis of variance (ANOVA) relies heavily on F-tests.
Beta distribution - Defined on the interval [0, 1], making it useful for modeling proportions and probabilities. Bayesian statistics often uses beta distributions as prior distributions.
Gamma distribution - A flexible two-parameter family that includes the exponential distribution as a special case. Insurance companies use gamma distributions to model claim sizes.
Log-normal distribution - When the logarithm of a variable follows a normal distribution, the variable itself follows a log-normal distribution. Stock prices, income distributions, and particle sizes often approximate log-normality.
Applications in statistics
Continuous distributions underpin most statistical methodology:
Inference and estimation - Confidence intervals and hypothesis tests assume particular distributions for sample statistics. The choice of distribution depends on sample size, whether variance is known, and other factors. Normal, t, chi-squared, and F distributions appear constantly in inferential statistics.
Regression analysis - Linear regression assumes that errors follow a normal distribution with constant variance. Violations of this assumption may require transformations or alternative methods like robust regression.
Quality control - Manufacturing processes use control charts based on normal distribution theory. Points falling outside control limits signal potential problems requiring investigation. Six Sigma methods rely heavily on normal distribution properties.
Risk management - Financial institutions model investment returns, credit defaults, and operational losses using continuous distributions. Value at Risk (VaR) calculations estimate the probability of losses exceeding specified thresholds.
Survival analysis - Medical researchers use exponential, Weibull, and other distributions to model time until death or disease recurrence. These methods inform treatment decisions and clinical trial design.
Simulation and Monte Carlo methods - Computer simulations generate random values from specified distributions to model complex systems. Financial pricing models, logistics optimization, and scientific experiments all employ such techniques.
Contrasts with discrete distributions
Continuous distributions differ fundamentally from discrete distributions in several ways:
Discrete variables take specific countable values. The number of defects in a product, customer arrivals per hour, and survey responses on a five-point scale are discrete. Probabilities can be assigned to individual values.
Continuous variables can take any value in an interval. Time measurements, physical dimensions, and monetary amounts (in principle) are continuous. Individual values have zero probability; only intervals have positive probability.
The Poisson distribution describes counts of rare events. The binomial distribution models successes in fixed numbers of trials. These discrete distributions have continuous counterparts: Poisson is approximated by normal for large means; binomial is approximated by normal when both np and n(1-p) exceed about 5.
{{{Concept}}} Primary topic {{{list1}}} Related topics {{{list2}}} Methods and techniques {{{list3}}}
See also
References
- Stigler, S.M. (1986). The History of Statistics: The Measurement of Uncertainty before 1900. Harvard University Press.
- Casella, G., & Berger, R.L. (2002). Statistical Inference. Duxbury Press.
- Johnson, N.L., Kotz, S., & Balakrishnan, N. (1994). Continuous Univariate Distributions. Wiley.
- Rice, J.A. (2007). Mathematical Statistics and Data Analysis. Thomson Brooks/Cole.
- Hald, A. (1998). A History of Mathematical Statistics from 1750 to 1930. Wiley.
Footnotes
<references> [1] A continuous random variable can take any value within an interval, typically resulting from measurements of physical quantities. [2] Laplace (1774) studied the mathematical properties of the normal distribution, though through historical error, credit often went primarily to Gauss who first referenced it in an 1809 paper. [3] The graph of a continuous probability distribution is a curve called the probability density function, where probability is represented by area under the curve. </references>