Random Quantile Experiment
Description
This applet simulates values from a random variable with a specified special distribution by computing a random quantile. The quantile function of the special distribution is \(F^{-1}\), where \(F\) is the distribution function. If \(U\) has the standard uniform distribution (the uniform distribution on the interval \((0, 1)\)), then \(X = F^{-1}(U)\) has the given special distribution. The graph on the left shows the distribution function/quantile function of the specified distribution, while the graph on the right shows the probability density function. On each run, the value of \(U\) is shown as the horizontal line in the left graph while the value of \(X\) is showns as the vertical line. The values of \(U\) and \(X\) are recorded in the table on the left. As the experiment runs, the empirical density and moments of \(X\) are shown in the right graph and recorded in the right table. the following distributions can be chosen with the selection box:
- The arcsine distribution on the interval \( (a, a + w) \).
- The discrete arcsine distribution on \( \{0, 1, \ldots, n\} \)
- The Benford's first digit distribution with base \( b \)
- The Benford mantissa distribution with base \( b \)
- The beta distribution with left shape parameter \(a\) and right shape parameter \(b\)
- The beta-binomial distribution with left beta parameter \( a \), right beta parameter \( b \), and \( n \) trials
- The beta-negative binomial distribution with left beta parameter \( a \), right beta parameter \( b \), and \( k \) successes
- The beta prime distribution with shape parameters \( a \) and \( b \)
- The binomial distribution with parameters \( n \) and \( p \)
- The birthday distribution with population size \( m \) and sample size \( n \)
- The Cauchy distribution with location parameter \( a \) and scale parameter \( b \)
- The chi-square distribution with \(n\) degrees of freedom
- The coupon collector distribution with population size \( m \) and distinct size \( k \)
- The exponential distribution with scale parameter \(b\)
- The exponential-logarithmic distribution with shape parameter \( p \) and scale parameter \( b \)
- The extreme value distribution with location parameter \( \mu \) and scale parameter \( \sigma \)
- The finite order statistic distribution with population size \( m \), sample size \( n \), and order \( k \)
- The Fisher \(F\) distribution with \(n\) degrees of freedom in the numerator and \(d\) degrees of freedom in the denominator
- The folded-normal distribution with parameters \( \mu \) and \( \sigma \)
- The gamma distribution with shape parameter \(k\) and scale parameter \(b\)
- The geometric distribution with success parameter \(p\)
- The Gompertz distribution with shape parameter \(a\) and scale parameter \( b \)
- The hypergeometric distribution with parameters \( m \), \( r \), and \( n \)
- The hyperbolic secant distribution with location parameter \( \mu \) and scale parameter \( \sigma \)
- The Irwin-Hall distribution with \( n \) terms
- The Laplace distribution with location parameter \( a \) and scale parameter \( b \)
- The Lévy distribution with location parameter \( a \) and scale parameter \( b \)
- The logarithmic series distribution with shape parameter \( p \)
- The logistic distribution with location parameter \(a\) and scale parameter \(b\)
- The log-logistic distribution with scale parameter \( b \) and shape parameter \( k \)
- The lognormal distribution with parameters \(\mu\) and \(\sigma\)
- The matching distribution with parameter \( n \)
- The Maxwell distribution with scale parameter \( b \)
- The negative binomial distribution with parameters \( k \) and \( p \)
- The normal distribution with mean \(\mu\) and standard deviation \(\sigma\)
- The Pareto distribution with shape parameter \(k\) and scale parameter \(b\)
- The Poisson distribution with parameter \( \lambda \)
- The Pólya distribution with parameters \( a \), \( b \), \( c \), and \( n \)
- The Rayleigh distribution with scale parameter \(b\)
- The semicircle distribution with center \(a\) and radius \( r \).
- The student \(t\) distribution with \(n\) degrees of freedom
- The triangle distribution with location parameter \( a \), scale parameter \(w \) and shape parameter \(p\).
- The uniform distribution on the interval \( [a, a + w] \)
- The discrete uniform distribution with \( n \) points, starting at \( a \), with step size \( h \).
- The U-Power distribution with shape parameter \( k \), location parameter \( \mu \), and scale parameter \( c \)
- The Weibull distribution with shape parameter \(k\) and scale parameter \(b\)
- The zeta distribution with shape parameter \( a \)
In each case, the parameters can be set with the input controls.