Merge pull request #115 from StatProofBook/master

update to master

Author: JoramSoch

I/PbA.md

@@ -8,7 +8,7 @@ title: "Proof by Author"
 
 - [Covariance matrix of the multinomial distribution](/P/mult-cov)
 
-### JoramSoch (390 proofs)
+### JoramSoch (391 proofs)
 
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
@@ -138,6 +138,7 @@ title: "Proof by Author"
 - [Invariance of the variance under addition of a constant](/P/var-inv)
 - [Inverse transformation method using cumulative distribution function](/P/cdf-itm)
 - [Joint likelihood is the product of likelihood function and prior density](/P/jl-lfnprior)
+- [Kullback-Leibler divergence for the Bernoulli distribution](/P/bern-kl)
 - [Kullback-Leibler divergence for the Dirichlet distribution](/P/dir-kl)
 - [Kullback-Leibler divergence for the gamma distribution](/P/gam-kl)
 - [Kullback-Leibler divergence for the matrix-normal distribution](/P/matn-kl)

I/PbN.md

@@ -424,3 +424,4 @@ title: "Proof by Number"
 | P416 | bvn-pdf | [Probability density function of the bivariate normal distribution](/P/bvn-pdf) | JoramSoch | 2023-09-22 |
 | P417 | bvn-pdfcorr | [Probability density function of the bivariate normal distribution in terms of correlation coefficient](/P/bvn-pdfcorr) | JoramSoch | 2023-09-29 |
 | P418 | mlr-olstr | [Ordinary least squares for multiple linear regression with two regressors](/P/mlr-olstr) | JoramSoch | 2023-10-06 |
+| P419 | bern-kl | [Kullback-Leibler divergence for the Bernoulli distribution](/P/bern-kl) | JoramSoch | 2023-10-13 |

I/PbT.md

@@ -165,6 +165,7 @@ title: "Proof by Topic"
 
 ### K
 
+- [Kullback-Leibler divergence for the Bernoulli distribution](/P/bern-kl)
 - [Kullback-Leibler divergence for the Dirichlet distribution](/P/dir-kl)
 - [Kullback-Leibler divergence for the gamma distribution](/P/gam-kl)
 - [Kullback-Leibler divergence for the matrix-normal distribution](/P/matn-kl)

I/PwS.md

@@ -45,6 +45,7 @@ title: "Proofs without Source"
 - [First raw moment is mean](/P/momraw-1st)
 - [Gamma distribution is a special case of Wishart distribution](/P/gam-wish)
 - [Joint likelihood is the product of likelihood function and prior density](/P/jl-lfnprior)
+- [Kullback-Leibler divergence for the Bernoulli distribution](/P/bern-kl)
 - [Kullback-Leibler divergence for the matrix-normal distribution](/P/matn-kl)
 - [Kullback-Leibler divergence for the normal distribution](/P/norm-kl)
 - [Linear combination of independent normal random variables](/P/norm-lincomb)

P/mlr-f.md

@@ -171,7 +171,7 @@ $$
 
 and under the [null hypothesis](/D/h0) \eqref{eq:mlr-f-h0}, it can be evaluated as:
 
-$$ \label{eq:mlr-t-s2}
+$$ \label{eq:mlr-f-s2}
 \begin{split}
 F &\overset{\eqref{eq:mlr-f-s1}}{=} \left( \hat{\gamma} -  C^\mathrm{T} \beta \right)^\mathrm{T} \left( \frac{n-p}{\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon}} Q \right) \left( \hat{\gamma} -  C^\mathrm{T} \beta \right) / q \\
 &\overset{\eqref{eq:mlr-f-h0}}{=} \hat{\gamma}^\mathrm{T} \left( \frac{n-p}{\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon}} Q \right) \hat{\gamma} / q \\
@@ -181,4 +181,6 @@ F &\overset{\eqref{eq:mlr-f-s1}}{=} \left( \hat{\gamma} -  C^\mathrm{T} \beta \r
 &\overset{\eqref{eq:mlr-est}}{=} \hat{\beta}^\mathrm{T} C \left( \frac{1}{\hat{\sigma}^2} \left( C^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} C \right)^{-1} \right) C^\mathrm{T} \hat{\beta} / q \\
 &= \hat{\beta}^\mathrm{T} C \left( \hat{\sigma}^2 C^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} C \right)^{-1} C^\mathrm{T} \hat{\beta} / q \; .
 \end{split}
-$$
+$$
+
+This means that the [null hypothesis](/D/h0) in \eqref{eq:mlr-f-h0} can be rejected when $F$ from \eqref{eq:mlr-f-s2} is as extreme or more extreme than the [critical value](/D/cval) obtained from [Fisher's F-distribution](/D/f) with $q$ numerator and $n-p$ denominator [degrees of freedom](/D/dof) using a [significance level](/D/alpha) $\alpha$.

P/mlr-t.md

@@ -161,4 +161,6 @@ t &\overset{\eqref{eq:mlr-t-s1}}{=} \frac{z}{\sqrt{v/(n-p)}} \\
 &\overset{\eqref{eq:mlr-rss}}{=} \frac{c^\mathrm{T} \hat{\beta}}{\sqrt{\frac{(y-X\hat{\beta})^\mathrm{T} V^{-1} (y-X\hat{\beta})}{n-p} \cdot c^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} c}} \\
 &\overset{\eqref{eq:mlr-est}}{=} \frac{c^\mathrm{T} \hat{\beta}}{\sqrt{\hat{\sigma}^2 c^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} c}} \; .
 \end{split}
-$$
+$$
+
+This means that the [null hypothesis](/D/h0) in \eqref{eq:mlr-t-h0} can be rejected when $t$ from \eqref{eq:mlr-t-s2} is as extreme or more extreme than the [critical value](/D/cval) obtained from [Student's t-distribution](/D/t) with $n-p$ [degrees of freedom](/D/dof) using a [significance level](/D/alpha) $\alpha$.