Skip to main content

t distribution

t Distribution

Probability Density FunctionThe formula for the probability density function of the t distribution isf(x) = (1 + x**2/nu)**(-(nu+1)/2)/[B(0.5,0.5*nu)*SQRT(nu)]
where B is the beta function and nu is a positive integer shape parameter. The formula for the beta function is
B(alpha,beta) = INTEGRAL(t**(alpha-1)*(1-t)**(beta-1)dt) where the integration is from 0 to 1
In a testing context, the t distribution is treated as a "standardized distribution" (i.e., no location or scale parameters). However, in a distributional modeling context (as with other probability distributions), the t distribution itself can be transformed with alocation parameter, mu, and a scale parameter, sigma.
The following is the plot of the t probability density function for 4 different values of the shape parameter.
plot of the t probability density function for 4 different values of the shape parameter
These plots all have a similar shape. The difference is in the heaviness of the tails. In fact, the t distribution with nu equal to 1 is aCauchy distribution. The t distribution approaches a normaldistribution as nu becomes large. The approximation is quite good for values of nu > 30.
Cumulative Distribution FunctionThe formula for the cumulative distribution function of the tdistribution is complicated and is not included here. It is given in theEvans, Hastings, and Peacock book.The following are the plots of the t cumulative distribution function with the same values of nu as the pdf plots above.
plot of the t cumulative distribution function with the same values of nu as the pdf plots above
Percent Point FunctionThe formula for the percent point function of the t distribution does not exist in a simple closed form. It is computed numerically.The following are the plots of the t percent point function with the same values of nu as the pdf plots above.
plot of the t percent point function with the same values of nu as the pdf plots above
Other Probability FunctionsSince the t distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit the formulas and plots for the hazard, cumulative hazard, survival, and inverse survival probability functions.
Common Statistics
Mean0 (It is undefined for nu equal to 1.)
Median0
Mode0
RangeInfinity in both directions.
Standard DeviationSQRT(nu/(nu-2))It is undefined for nu equal to 1 or 2.
Coefficient of VariationUndefined
Skewness0. It is undefined for nu less than or equal to 3. However, the t distribution is symmetric in all cases.
Kurtosis3*(nu-2)/(nu-4)It is undefined for nu less than or equal to 4.
Parameter EstimationSince the t distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit any discussion of parameter estimation.
CommentsThe t distribution is used in many cases for the critical regions for hypothesis tests and in determining confidence intervals. The most common example is testing if data are consistent with the assumed process mean.
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the t distribution.

Comments

Post a Comment

Popular posts from this blog

Methods of correlation

Correlation Methods  Correlation and Covariance Matrices Fisher's  z  Transformation ( z r ) Pearson Product-Moment Correlation Coefficient (Pearson's  r ) Spearman's Rank-Order Correlation Coefficient (Spearman's ρ) Kendall's Rank Correlation Coefficient (Kendall's τ) Correlation and Covariance Matrices You can generate a correlation or covariance matrix from numeric data columns, and have the choice of storing the computation results in an-autogenerated worksheet, or display the results in a table format whose values can be color coded. This method requires multiple numeric data columns whose values should be stored in a single worksheet. An example of a correlation matrix displayed as a color-coded table is shown below. Using Fisher's  z  Transformation ( z r ) This option is provided to allow transforming a skewed sampling distribution into a normalized format. The theoretical sampling distribution of the corr...
2 comments
Read more

The most femiliar statisticians

Gertrude Cox :   Gertrude Mary Cox (of Experimental Statistics at North Carolina State University. She was later appointed director of both the Institute of Statistics of 1900 - 1978) was an influential American statistician and founder of the department the Consolidated University of North Carolina and the Statistics Research Division of North Carolina State University. Her most important and influential research dealt with experimental design; she wrote an important book on the subject with W. G. Cochran. In 1949 Cox became the first female elected into the International Statistical Institute and in 1956 she was president of the American Statistical Association. From 1931 to 1933 Cox undertook graduate studies in statistics at the  University of California at Berkeley , then returned to Iowa State College as assistant in the Statistical Laboratory. Here she worked on the  design of experiments . In 1939 she was appointed assistant professor of statisti...
Post a Comment
Read more

Cauchy distribution

Cauchy Distribution Probability Density Function The general formula for the  probability density function  of the Cauchy distribution is where  t  is the  location parameter  and  s  is the  scale parameter . The case where  t  = 0 and  s  = 1 is called the  standard Cauchy distribution . The equation for the standard Cauchy distribution reduces to Since the general form of probability functions can be  expressed in terms of the standard distribution , all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the standard Cauchy probability density function. Cumulative Distribution Function The formula for the  cumulative distribution function  for the Cauchy distribution is The following is the plot of the Cauchy cumulative distribution function. Percent Point Function The formula for the  percent point function  of...
Post a Comment
Read more