equation_確率分布

 

 

\( C_{ p } = \displaystyle\frac{ S_{ U } – S_{ L } }{ 6s } \)

\( k = \displaystyle\frac{ | ( S_{ U } + S_{ L } )/2 – \bar{ X } | }{ ( S_{ U } – S_{ L } )/2 } \)

\( k < 1 ならば C_{ pk } = ( 1 – k ) \times C_{ p } \)

\( k \geq 1 ならば C_{ pk } = 0 \)

\( C_{ pk } = \displaystyle\min  ( \displaystyle\frac{ S_{ U } – \bar{ X }}{ 3s} , \displaystyle\frac{ \bar{ X } – S_{ L } }{ 3s } ) \)

\( C_{ p } = \displaystyle\frac{ S_{ U } – \bar{ X } }{ 3s } \)

\( C_{ p } = \displaystyle\frac{ \bar{ X } – S_{ L } }{ 3s } \)

 

\( y = f(x) \)   \( y = f(x) \)

\( f(x) \geq 0 \)

\( Pr( a \lt x \leq b ) = \displaystyle \int_a^b f(x) dx \)

\( ( a ,  b ] ( a \lt x \leq b )\)   \( Pr( a \lt x \leq b ) \)

\( f(x) \)   \( f(x) \)

\( Pr( – \infty \lt x \leq \infty ) = \displaystyle \int_{-\infty}^{ \infty } f(x) dx \)

\( Pr( – \infty \lt x \leq \infty ) = \displaystyle \int_{-\infty}^{ \infty } f(x) dx = 1 \)

\( E(X) = \mu \)   \( V(X) \)   \( D(X) \)   \( \sigma \)

\( f(x) \)   \( f(x) \)

\( E(X) = \displaystyle \int_{}^{} x f(x) dx \) 

\( V(X) = \displaystyle \int_{}^{} ( x – \mu )^2 f(x) dx \) 

\( \bar{ X } \)

\( E( aX ) = a E(X) \)

\( E( a_{ 1 } X_{ 1 } + a_{ 2 } X_{ 2 } +  \cdots + a_{ n } X_{ n } ) = a_{ 1 }E(X_{ 1 }) + a_{ 2 }E(X_{ 2 }) + \cdots + a_{ n }E(X_{ n }) \)

\( E( X_{ 1 } + X_{ 2 } ) = E(X_{ 1 }) + E(X_{ 2 }) \)

\( E( X_{ 1 } – X_{ 2 } ) = E(X_{ 1 }) – E(X_{ 2 }) \)

\( V( aX ) = a^2 V(X) \)

\( V( a_{ 1 } X_{ 1 } + a_{ 2 } X_{ 2 } +  \cdots + a_{ n } X_{ n } ) = a_{ 1 }^2 V(X_{ 1 }) + a_{ 2 }^2 V(X_{ 2 }) + \cdots + a_{ n }^2 V(X_{ n }) \)

\( V( X_{ 1 } + X_{ 2 } ) = V(X_{ 1 }) + V(X_{ 2 }) \)

\( V( X_{ 1 } – X_{ 2 } ) = V(X_{ 1 }) + V(X_{ 2 }) \)

\( V( X_{ 1 } + X_{ 2 } ) = V(X_{ 1 }) + V(X_{ 2 }) + 2 Cov( X_{ 1 } , X_{ 2 }) \)

\( V( X_{ 1 } – X_{ 2 } ) = V(X_{ 1 }) + V(X_{ 2 }) – 2 Cov( X_{ 1 } , X_{ 2 })\)

\( 2 Cov( X_{ 1 } , X_{ 2 }) = E[ \{ X_{ 1 } – E( X_{ 1 } ) \} \{ X_{ 2 } – E( X_{ 2 } ) ] \)

\( Cov( X_{ 1 } , X_{ 2 }) = E[ \{ X_{ 1 } – E( X_{ 1 } ) \} \{ X_{ 2 } – E( X_{ 2 } ) ] \)

  \( f(x) = \displaystyle\frac{ 1 }{ \sqrt{ 2 \pi \sigma } } \exp \left\{ – \frac{ 1 }{ 2 } \left( \displaystyle\frac{ x – \mu }{ \sigma } \right)^2 \right\} \) 

\( N( \mu , \sigma^2 ) \)   \( E(X) = \mu \)   \( D(X) = \sigma   ( V(X) = \sigma^2 )  \)

\( \mu = 0 \)   \( \sigma^2 = 1 \)   \( N( 0 , 1^2 ) \) 

\( Pr( X = x ) \)    \( Pr( X = x ) = {}_n \mathrm{ C }_x P^{ x } ( 1 – P)^{ n – x } = \displaystyle\frac{ n! }{ x! ( n – x )! } P^{ x } ( 1 – P )^{ n – x } \)

\( {}_n \mathrm{ C }_x \)   \( {}_n \mathrm{ C }_x = \displaystyle\frac{ n! }{ x! ( n – x )! } \)

\( E( X ) = nP \)    \( D( X ) = \sqrt{ n P ( 1 – P ) } \) 

\( p = X/n \)   \( E( p ) = P \)   \( D( p ) = \sqrt{ \displaystyle\frac{ P ( 1 -P ) }{ n } } \)

\( nP \geq 5 ,   n ( 1-P ) \geq 5 \)

\( np = \lambda \)

\( Pr( X = k ) = \displaystyle\frac{ e^{ – \lambda } \lambda ^k }{ k! }   ( k = 0, 1, 2, \cdots  ) \)

\( E( k ) = \lambda \)   \( D( k ) = \sqrt{ \lambda } \)

 

\( Pr( u \geq K_{ p } ) = P \)

\( U = \displaystyle\frac{ X  – \mu }{ \sigma } \)

\( N( 20, 4^2 ) \)

\( U = \displaystyle\frac{ 24  – 20 }{ 4 } = 1 \)

\( N( 100, 30^2 ) \)

\( U = \displaystyle\frac{ 53.5  – 100 }{ 30 } = 1.55 \)

 

\( S(xy) = \displaystyle \sum_{ i=1 }^n ( x_{ i } – \bar{ x })( y_{ i } – \bar{ y }) \)

\( S(xy) = \displaystyle \sum_{ i=1 }^n  x_{ i }y_{ i } – n \sum_{ i=1 }^n x_{ i } \sum_{ i=1 }^n y_{ i } \)

\( S(xy) = \displaystyle \sum_{ i=1 }^n  x_{ i }y_{ i } – \displaystyle\frac{\sum_{ i=1 }^n x_{ i } \sum_{ i=1 }^n y_{ i } }{ n } \)

\( S(xy) = – \displaystyle\frac{\sum_{ i=1 }^n x_{ i } \sum_{ i=1 }^n y_{ i } }{ n } \)