F distribution

Examples of F distribution in the following topics:

• Notation

  • The notation for the F distribution is F∼Fdf(num),df(denom) where df(num) = dfbetween and df(denom) = dfwithin
  • The mean for the F distribution is $\mu =\frac{ df(num) }{df(denom)1 }$

• The F-Test

  • An F-test is any statistical test in which the test statistic has an F-distribution under the null hypothesis.
  • An F-test is any statistical test in which the test statistic has an F-distribution under the null hypothesis.
  • (Note that all populations involved must be assumed to be normally distributed.)
  • The F-distribution exhibits the following properties, as illustrated in the above graph:
  • The F-distribution is skewed to the right and begins at the x-axis, meaning that F-values are always positive.

• Summary

  • The distribution for the test is the F distribution with 2 different degrees of freedom.
  • The distribution for the hypothesis test is the F distribution with 2 different degrees of freedom.
  • The populations from which the two samples are drawn are normally distributed.

• Variance Estimates

  • An $F$-test for the null hypothesis that two normal populations have the same variance is sometimes used; although, it needs to be used with caution as it can be sensitive to the assumption that the variables have this distribution.
  • This particular situation is of importance in mathematical statistics since it provides a basic exemplar case in which the $F$ distribution can be derived.
  • Let $X_1, \dots, X_n$ and $Y_1, \dots, Y_m$ be independent and identically distributed samples from two populations which each have a normal distribution.
  • It has an $F$-distribution with $n-1$ and $m-1$ degrees of freedom if the null hypothesis of equality of variances is true.
  • These $F$-tests are generally not robust when there are violations of the assumption that each population follows the normal distribution, particularly for small alpha levels and unbalanced layouts.

• Analysis of variance (ANOVA) and the F test

  • A p-value can be computed from the F statistic using an F distribution, which has two associated parameters: df1 and df2.
  • An F distribution with 3 and 323 degrees of freedom, corresponding to the F statistic for the baseball hypothesis test, is shown in Figure 5.29.
  • If H0 is true and the model assumptions are satisfied, the statistic F follows an F distribution with parameters df1 = k−1 and df2 = n−k.
  • The upper tail of the F distribution is used to represent the p-value.
  • An F distribution with df1 = 3 and df2 = 323.

• Speed Distribution of Molecules

  • A gas of many molecules has a predictable distribution of molecular speeds, known as the Maxwell-Boltzmann distribution.
  • Maxwell-Boltzmann distribution is a probability distribution.
  • $f_\mathbf{v} (v_x, v_y, v_z) = \left(\frac{m}{2 \pi kT} \right)^{3/2} \exp \left[- \frac{m(v_x^2 + v_y^2 + v_z^2)}{2kT} \right]$,
  • The Maxwell–Boltzmann distribution for the speed follows immediately from the distribution of the velocity vector, above.
  • $f(v) = \sqrt{\left(\frac{m}{2 \pi kT}\right)^3}\, 4\pi v^2 \exp \left(\frac{-mv^2}{2kT}\right)$ for speed v.

• The One-Way F-Test

  • This $F$-statistic follows the $F$-distribution with $K-1$, $N-K$ degrees of freedom under the null hypothesis.
  • The data were distributed as follows:
  • Remember that the null hypothesis claims that the sorority groups are from the same normal distribution.
  • The alternate hypothesis says that at least two of the sorority groups come from populations with different normal distributions.
  • This chart shows example p-values for two F-statistics: p = 0.05 for F = 3.68, and p = 0.00239 for F = 9.27.

• Continuous Probability Distributions

  • A continuous probability distribution is a probability distribution that has a probability density function.
  • This requirement is stronger than simple continuity of the cumulative distribution function, and there is a special class of distributions—singular distributions, which are neither continuous nor discrete nor a mixture of those.
  • An example is given by the Cantor distribution.
  • For example, the uniform distribution on the interval $\left[0, \frac{1}{2}\right]$ has probability density $f(x) = 2$ for $0 \leq x \leq \frac{1}{2}$ and $f(x) = 0$ elsewhere.
  • The standard normal distribution has probability density function:

• Inverse Compton Spectra - Single Scattering

  • Let's suppose that we have an isotropic distribution of photons of a single energy $E_0$ and a beam of electrons traveling along the $x$-axis with energy $\gamma m c^2$ and density $N$.
  • Let's assume that there are many beams isotropically distributed, so we need to find the mean value of $j(E_f,\mu_f)$ over angle,
  • so we find that the scattered photons have an energy distribution $E^{-s}$ where $s=(p-1)/2$.
  • This power-law distribution is valid over a limited range of photon energies.
  • If the initial photon distribution peaks at ${\bar E}$ the power-law will work between $4 \gamma_1^2 {\bar E}$ and $4 \gamma_2^2 {\bar E}

• Introduction to the Fourier Integral

  • $\displaystyle{f _1 = \frac{1}{2l}, \hspace{2mm} f _2 = \frac{2}{2l}, \hspace{2mm} f _3 = \frac{3}{2l}, \cdots , \hspace{2mm} f _k = \frac{k}{2l} \cdots }$
  • $ \displaystyle{f _1 = \frac{1}{2L}, \hspace{2mm} f _2 = \frac{2}{2L}, \hspace{2mm} f _3 = \frac{3}{2L}, \cdots , \hspace{2mm} f _k = \frac{k}{2L} \cdots }$
  • The Fourier coefficients become more and more densely distributed, until, in the limit that $L \rightarrow \infty$ , the coefficient sequence $c_n$ becomes a continuous function.
  • A function $f(t)$ is related to its Fourier transform $f(\omega)$ via:
  • $\displaystyle{g(t) = \int _{-\infty} ^{\infty} G(f) e^{2\pi i f t} ~df}$