One can show fairly easily that a simple rational function of a random variable with an F-distribution actually has a Beta distribution. The F-distribution with $\nu$ and $\xi$ degrees of freedom is $F=(\chi^2_\nu/\nu)/(\chi^2_\xi/\xi)$, where the two chi-square random variables are independent. As the degrees of freedom increase, the chi-square. The F-distribution is one of the great work-horses of applied statistics. When the degrees of freedom are greater than or equal to 2, the maximum value for Y occurs when 2 - 2. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution. This comes up when one thinks about the F-distribution (The "F" stands for "Fisher", as in Ronald Aylmer Fisher, one of the most famous 20th-century scientists). A chi-squared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. If you find the probability that that random variable is $<1/2$, you'll get a far bigger number with a $\chi^2_1$ than with $\chi^2_/(2n)$.ĭividing the degrees of freedom by the chi-square random variable results in a distribution of quite a different shape, not merely a rescaled chi-square distribution. Johan Stax Jakobsen at 10:49 oh.so they are the same. There is no contradiction between the two statements. The expected value does become the same as that of a $\chi^2_1$ distribution, but the shape of the density function is quite different. One is chi square distributed with k degrees of freedom and the other with n -1.
Dividing a chi-square-distributed random variable by its degrees of freedom is merely rescaling it doesn't change the shape parameter in the gamma distribution.
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