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StudentTDistribution

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StudentTDistribution[Nu]
represents a Student t distribution with Nu degrees of freedom.
StudentTDistribution[Mu, Sigma, Nu]
represents a Student t distribution with location parameter Mu, scale parameter Sigma and Nu degrees of freedom.
  • The probability density for value x in a Student t distribution with Nu degrees of freedom is proportional to (nu/(nu+x^2))^((1+nu)/2). »
  • With location parameter Mu and scale Sigma, (x-mu)/sigma follows a standard Student t distribution with Nu degrees of freedom.
  • For integer Nu, the Student t distribution gives the distribution of the deviation from the true mean of the observed mean for a sample of Nu values from a normal distribution, normalized by standard deviation of the sample.
EXAMPLESCLOSE ALL
Basic Examples   (2)
The mean and variance of a Student t distribution:
The probability density function depends on the Beta function:
The mean and variance of a Student t distribution:
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The probability density function depends on the Beta function:
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Scope   (3)
Generate a set of random numbers that have the Student t distribution:
Properties based on higher-order moments:
The 0.75 quantile of a Student t distribution with Nu=5:
Applications   (3)
Compute p-values for a t-test with n degrees of freedom and alternative hypothesis X<t:
Alternative hypothesis X>t:
Alternative hypothesis |X|>t:
Plot the cumulative distribution function of the random variable:
A contour plot as both x and Nu are varied:
Properties & Relations   (6)
The probability density function integrates to unity:
A Student t distribution with Nu degrees of freedom has finite moments only when -1<k<nu:
StudentTDistribution[Nu] has location mu=0 and scale sigma=1:
The two forms are related by a change of variable:
StudentTDistribution[1] is equivalent to CauchyDistribution[0, 1]:
The Student t distribution converges to the standard normal distribution as Nu tends to infinity:
Possible Issues   (2)
StudentTDistribution is not defined when Nu is a not a positive real number:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
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