LaTeX/Mode Mix Example

Combining In-Line and Multi-Line Formats

Let $\mu$ and $\sigma$ be the population mean and standard deviation. Fix the sample size $n$ . Let $\bar X$ be the sample mean, $S^2$ the sample variance given by

$S^2 = {1 \over n-1}\sum (X_i-\bar X)^2$

It is elementary to show $E(\bar X)=\mu$ . We now show $E(S^2)=\sigma^2$ .

$\begin{eqnarray*} \sum (X_i-\bar X)^2 & = & \sum ((X_i-\mu)-(\bar X -\mu))^2 \\ & = & \sum ((X_i-\mu)^2) -n(\bar X-\mu)^2 , \end{eqnarray*}$

since $\sum (X_i-\mu)=(\sum X_i) - n\mu = n(\bar X-\mu)$ .

Then

$\begin{eqnarray*} E(S^2) & = & E({1 \over n-1}\sum (X_i-\bar X)^2) \\ & = & {1 \over n-1}( \sum E((X_i-\mu)^2)-nE((\bar X-\mu)^2) ) \\ & = & {1 \over n-1}( \sum \sigma^2(X_i)- n\sigma^2(\bar X) ) \end{eqnarray*}$

But $\sigma^2(X_i)=\sigma^2$ , $\sigma^2(\bar X)=\sigma^2/n$ , so

$E(S^2) = {1 \over n-1}(n\sigma^2 -n(\sigma^2/n))=\sigma^2 .$

Combining LaTeX and Code Blocks

  NB. u adjusts deviation population size when computing variance
  NB. y is population
  NB. x is the sample size
  mean=: +/ % #
  s0=: %:@((%u)~@# * mean)@:*:@:- mean NB. "sample standard deviation"
  s1=: %:@mean@:*:@:- mean              NB. "population standard deviation"
  n=:x
  n1=:u n

  mu=: mean y =.x:y

  all possible samples (each has equal probability)
  possibilities=:y {~"1 (n##y)#:i.n^~#y

$\sum (X_i-\bar X)^2$

  t0=: ([:+/2^~]-mean)"1 possibilities

$=\sum ((X_i-\mu)-(\bar X -\mu))^2$

  t1=: ([:+/2^~(mu-~])-mu-~mean)"1 possibilities
  assert.t0-:t1

$=\sum ((X_i-\mu)^2) -n(\bar X-\mu)^2$ ,

  t2=: (([:+/2^~mu-~])-n*2^~mu-~mean)"1 possibilities
  assert.t0-:t2

since $\sum (X_i-\mu)=(\sum X_i) - n\mu = n(\bar X-\mu)$ .

  assert. 1=#~.(([:+/mu-~])"1;((n*mu)-~+/)"1;(n*mu-~mean)"1) possibilities

Then $E(S^2)=E((1/n-1)\sum (X_i-\bar X)^2$

  t3=: mean *:@s0"1 possibilities
  t4=: mean ((%n1)*[:+/2^~]-mean)"1 possibilities
  assert. t3-:t4

$=(1/n-1)( \sum E((X_i-\mu)^2)-nE((\bar X-\mu)^2 ))$

  t5=:(%n1)* ( (+/n#mean *:mu-~y)) - n*([:mean ([:*:mu-~mean)"1) possibilities
  assert. t3-:t5

$=(1/n-1)( \sum \sigma^2(X_i)- n\sigma^2(\bar X) )$

  t6=:(%n1)* (+/*:n#s1 y) - n* *:@s1@:(mean"1) possibilities
  assert. t3-:t6

But $\sigma^2(X_i)=\sigma^2$ , $\sigma^2(\bar X)=\sigma^2/n$ , so

  assert. (*:s1 mean"1 possibilities) -: (%n) * *:s1 y

$E(S^2)=(1/n-1)(n\sigma^2 -n(\sigma^2/n))=\sigma^2$ .

  t7=: (%n1)* (n**:s1 y) - *:s1 y
  assert. t3-:t7