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    LN V (X) Derivation1

    Uploaded by

    hasanuzzaman
    The document presents the calculation of the variance V(X) of a normal distribution with mean μ and variance σ^2. It first calculates the expected value E(X^2) by integrating the probability density function from -∞ to ∞. Through algebraic manipulations, it simplifies this to e^{2σ^2}. It then uses this to derive an expression for the variance V(X) = E(X^2) - [E(X)]^2 in terms of μ and σ^2.

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    LN V (X) Derivation1

    Uploaded by

    hasanuzzaman
    18 views2 pages
    The document presents the calculation of the variance V(X) of a normal distribution with mean μ and variance σ^2. It first calculates the expected value E(X^2) by integrating the probability density function from -∞ to ∞. Through algebraic manipulations, it simplifies this to e^{2σ^2}. It then uses this to derive an expression for the variance V(X) = E(X^2) - [E(X)]^2 in terms of μ and σ^2.

    Original Description:

    LN derivation

    Original Title

    LN v(X) Derivation1

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    © © All Rights Reserved

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    The document presents the calculation of the variance V(X) of a normal distribution with mean μ and variance σ^2. It first calculates the expected value E(X^2) by integrating the probability density function from -∞ to ∞. Through algebraic manipulations, it simplifies this to e^{2σ^2}. It then uses this to derive an expression for the variance V(X) = E(X^2) - [E(X)]^2 in terms of μ and σ^2.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    18 views2 pages

    LN V (X) Derivation1

    Uploaded by

    hasanuzzaman
    The document presents the calculation of the variance V(X) of a normal distribution with mean μ and variance σ^2. It first calculates the expected value E(X^2) by integrating the probability density function from -∞ to ∞. Through algebraic manipulations, it simplifies this to e^{2σ^2}. It then uses this to derive an expression for the variance V(X) = E(X^2) - [E(X)]^2 in terms of μ and σ^2.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
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     

    V  X   E X 2  E  X 2

    Let   0 and E X 2   E X '

    x2

      '  2x
    e e 2 2
    E X2  dx
     2  2

    x2
     2 x
     2 2


    e
     dx
     2  2

     x 2 2 x.2 2 
     

     e  2 2 
       2  2
    dx

     x 2 2 x.2 2 2 2 4 2 


     
     e  2 2 
       2  2
    dx

      x 2 2 
     
     2
    2
     2
    2


    e e
     dx
     2  2

      x 2 2 
     
     2
    2
     


    e
     e 2
    2
    dx
     2  2

     e 2 .1
    2

     e 2
    2

    Therefore,

    x2

        e e 2x 2 2
    dx  e 2
    2
    E X 2 '
     2  2
    Again,


     x   2

        e e 2x 2 2
    E X2  dx
     2  2

    Let x    y then x  y   .

     dx  dy


     y 2
    e2 y   e
        2 2
    E X2  dx
     2  2


     y 2
     2 y 2 2 2


    e e e
     dx
     2  2

    y2

     2y 2 2


    e e
     e2 dx
     2  2

     e2  .E Y 2 '  
     e2  .e2
    2

     
    V  X   E X 2  E  X 2

    2 2
     
    2  2 2
    e e e 2 e 2

     e2 e2  e2 


    2 2

     e2  e  e2 
    2 2 2

     e2  e  1
    2
     2

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