Merge pull request #97 from StatProofBook/master

update to master

Author: JoramSoch

D/beta-data.md

@@ -9,7 +9,7 @@ date: 2020-06-28 21:16:00
 
 title: "Beta-distributed data"
 chapter: "Statistical Models"
-section: "Probability data"
+section: "Frequency data"
 topic: "Beta-distributed data"
 definition: "Definition"
 

D/bin-data.md

@@ -9,7 +9,7 @@ date: 2020-07-07 07:04:00
 
 title: "Binomial observations"
 chapter: "Statistical Models"
-section: "Categorical data"
+section: "Count data"
 topic: "Binomial observations"
 definition: "Definition"
 

D/dir-data.md

@@ -9,7 +9,7 @@ date: 2020-10-22 05:06:00
 
 title: "Dirichlet-distributed data"
 chapter: "Statistical Models"
-section: "Probability data"
+section: "Frequency data"
 topic: "Dirichlet-distributed data"
 definition: "Definition"
 

D/mult-data.md

@@ -9,7 +9,7 @@ date: 2020-07-07 07:12:00
 
 title: "Multinomial observations"
 chapter: "Statistical Models"
-section: "Categorical data"
+section: "Count data"
 topic: "Multinomial observations"
 definition: "Definition"
 

D/pgf.md

@@ -29,18 +29,18 @@ username: "JoramSoch"
 
 **Definition:**
 
-1) If $X$ is a discrete [random variable](/D/rvar) taking values in the non-negative integers $\left\lbrace 0, 1, \ldots \right\rbrace$, then the probability-generating function of $X$ is defined as
+1) If $X$ is a [discrete](/D/rvar-disc) [random variable](/D/rvar) taking values in the non-negative integers $\left\lbrace 0, 1, \ldots \right\rbrace$, then the probability-generating function of $X$ is defined as
 
 $$ \label{eq:pgf-var}
-G_X(z) = \mathrm{E} \left[ z^X \right] = \sum_{x=0}^{\infty} p(x) \, z^x
+G_X(z) = \sum_{x=0}^{\infty} p(x) \, z^x
 $$
 
 where $z \in \mathbb{C}$ and $p(x)$ is the [probability mass function](/D/pmf) of $X$.
 
-2) If $X$ is a discrete [random vector](/D/rvec) taking values in the $n$-dimensional integer lattice $x \in \left\lbrace 0, 1, \ldots \right\rbrace^n$, then the probability-generating function of $X$ is defined as
+2) If $X$ is a [discrete](/D/rvar-disc) [random vector](/D/rvec) taking values in the $n$-dimensional integer lattice $x \in \left\lbrace 0, 1, \ldots \right\rbrace^n$, then the probability-generating function of $X$ is defined as
 
 $$ \label{eq:cgf-vec}
-G_X(z) = \mathrm{E} \left[ {z_1}^{X_1} \cdot \ldots \cdot {z_n}^{X_n} \right] = \sum_{x_1=0}^{\infty} \cdots \sum_{x_n=0}^{\infty} p(x_1,\ldots,x_n) \, {z_1}^{x_1} \cdot \ldots \cdot {z_n}^{x_n}
+G_X(z) = \sum_{x_1=0}^{\infty} \cdots \sum_{x_n=0}^{\infty} p(x_1,\ldots,x_n) \, {z_1}^{x_1} \cdot \ldots \cdot {z_n}^{x_n}
 $$
 
 where $z \in \mathbb{C}^n$ and $p(x)$ is the [probability mass function](/D/pmf) of $X$.

D/poiss-data.md

@@ -9,7 +9,7 @@ date: 2020-03-22 22:50:00
 
 title: "Poisson-distributed data"
 chapter: "Statistical Models"
-section: "Poisson data"
+section: "Count data"
 topic: "Poisson-distributed data"
 definition: "Definition"
 

D/poissexp.md

@@ -9,7 +9,7 @@ date: 2020-03-22 22:57:00
 
 title: "Poisson distribution with exposure values"
 chapter: "Statistical Models"
-section: "Poisson data"
+section: "Count data"
 topic: "Poisson distribution with exposure values"
 definition: "Definition"
 

I/PbA.md

@@ -8,7 +8,7 @@ title: "Proof by Author"
 
 - [Covariance matrix of the multinomial distribution](/P/mult-cov)
 
-### JoramSoch (339 proofs)
+### JoramSoch (344 proofs)
 
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
@@ -276,6 +276,8 @@ title: "Proof by Author"
 - [Probability of the empty set](/P/prob-emp)
 - [Probability under mutual exclusivity](/P/prob-exc)
 - [Probability under statistical independence](/P/prob-ind)
+- [Probability-generating function is expectation of function of random variable](/P/pgf-mean)
+- [Probability-generating function of the binomial distribution](/P/bin-pgf)
 - [Projection matrix and residual-forming matrix are idempotent](/P/mlr-idem)
 - [Projection of a data point to the regression line](/P/slr-proj)
 - [Quantile function is inverse of strictly monotonically increasing cumulative distribution function](/P/qf-cdf)
@@ -321,6 +323,7 @@ title: "Proof by Author"
 - [Second central moment is variance](/P/momcent-2nd)
 - [Self-covariance equals variance](/P/cov-var)
 - [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr)
+- [Square of expectation of product is less than or equal to product of expectation of squares](/P/mean-prodsqr)
 - [Sums of squares for simple linear regression](/P/slr-sss)
 - [Symmetry of the covariance](/P/cov-symm)
 - [Symmetry of the covariance matrix](/P/covmat-symm)
@@ -334,6 +337,8 @@ title: "Proof by Author"
 - [Transposition of a matrix-normal random variable](/P/matn-trans)
 - [Two-sample t-test for independent observations](/P/ug-ttest2)
 - [Two-sample z-test for independent observations](/P/ugkv-ztest2)
+- [Value of the probability-generating function for argument one](/P/pgf-one)
+- [Value of the probability-generating function for argument zero](/P/pgf-zero)
 - [Variance of constant is zero](/P/var-const)
 - [Variance of parameter estimates for simple linear regression](/P/slr-olsvar)
 - [Variance of the Bernoulli distribution](/P/bern-var)

I/PbN.md

@@ -364,3 +364,8 @@ title: "Proof by Number"
 | P356 | beta-chi2 | [Relationship between chi-squared distribution and beta distribution](/P/beta-chi2) | JoramSoch | 2022-10-07 |
 | P357 | betabin-mome | [Method of moments for beta-binomial data](/P/betabin-mome) | JoramSoch | 2022-10-07 |
 | P358 | bin-margcond | [Marginal distribution of a conditional binomial distribution](/P/bin-margcond) | JoramSoch | 2022-10-07 |
+| P359 | mean-prodsqr | [Square of expectation of product is less than or equal to product of expectation of squares](/P/mean-prodsqr) | JoramSoch | 2022-10-11 |
+| P360 | pgf-mean | [Probability-generating function is expectation of function of random variable](/P/pgf-mean) | JoramSoch | 2022-10-11 |
+| P361 | pgf-zero | [Value of the probability-generating function for argument zero](/P/pgf-zero) | JoramSoch | 2022-10-11 |
+| P362 | pgf-one | [Value of the probability-generating function for argument one](/P/pgf-one) | JoramSoch | 2022-10-11 |
+| P363 | bin-pgf | [Probability-generating function of the binomial distribution](/P/bin-pgf) | JoramSoch | 2022-10-11 |

I/PbT.md

@@ -327,6 +327,8 @@ title: "Proof by Topic"
 - [Probability of the empty set](/P/prob-emp)
 - [Probability under mutual exclusivity](/P/prob-exc)
 - [Probability under statistical independence](/P/prob-ind)
+- [Probability-generating function is expectation of function of random variable](/P/pgf-mean)
+- [Probability-generating function of the binomial distribution](/P/bin-pgf)
 - [Projection matrix and residual-forming matrix are idempotent](/P/mlr-idem)
 - [Projection of a data point to the regression line](/P/slr-proj)
 - [Proof Template](/P/-temp-)
@@ -384,6 +386,7 @@ title: "Proof by Topic"
 - [Second central moment is variance](/P/momcent-2nd)
 - [Self-covariance equals variance](/P/cov-var)
 - [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr)
+- [Square of expectation of product is less than or equal to product of expectation of squares](/P/mean-prodsqr)
 - [Sums of squares for simple linear regression](/P/slr-sss)
 - [Symmetry of the covariance](/P/cov-symm)
 - [Symmetry of the covariance matrix](/P/covmat-symm)
@@ -404,6 +407,8 @@ title: "Proof by Topic"
 
 ### V
 
+- [Value of the probability-generating function for argument one](/P/pgf-one)
+- [Value of the probability-generating function for argument zero](/P/pgf-zero)
 - [Variance of constant is zero](/P/var-const)
 - [Variance of parameter estimates for simple linear regression](/P/slr-olsvar)
 - [Variance of the Bernoulli distribution](/P/bern-var)