Statistic

Maximum Likelihood

Bayesian Expectation Values

Symbol

Value

Symbol

Value

Limit n

location m

m ^

Y ¯

m ¯ = m | ( 2 ) ( n )

Y ¯

scale s

s ^

S Y 2 ¯ Y ¯ 2

s ¯ = s | ( 2 ) ( n )

( n 2 ) 1 2 Γ ( ( n 1 ) / 2 ) Γ ( n / 2 ) S

S

var ( m )

σ ^ m 2

S 2 / n

Δ m 2 = ( m m ¯ ) 2 | ( 2 ) ( n )

S 2 n 2

S 2 / n

var ( s )

σ ^ s 2

S 2 / 2 n

Δ s 2 = ( s s ¯ ) 2 | ( 2 ) ( n )

[ n n 2 n Γ ( ( n 1 ) / 2 ) 2 2 Γ ( n / 2 ) 2 ] S 2

S 2 / 2 n

Sample Statistics:

Y ¯ = 1 n k = 1 n ln ( z k )

Y 2 ¯ = 1 n k = 1 n ln ( z k ) 2