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Normal distribution (Single variable)

Where do you meet this distribution?

  • Standard score
  • Finance, Economics : changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal
  • Average of stochastic variables : Central Limit Theorem
  • Statistical mechanics : Velocities of the molecules in the ideal gas
  • Quantum physics : Probability density function of a ground state in a quantum harmonic oscillator
  • Error analysis

Shape of Distribution

Basic Properties

  • Two parameters m, \sigma are required.
    \sigma>0

    These parameters are Mean and Standard Deviation of the distribution respectively.

  • Continuous distribution defined on entire range
  • This distribution is always symmetric.

Probability

  • Cumulative distribution function
    F(x)=\int_{-\infty}^{x}\phi\left(\frac{t-m}{\sigma}\right)\text{d}t

    where \phi(\cdot) is Probability density function of standard normal distribution.

  • Probability density function
    f(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(x-m)^2}{2\sigma^2}\right]
  • How to compute these on Excel.
     
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    A B
    Data Description
    0.5 Value for which you want the distribution
    8 Value of parameter M
    2 Value of parameter Sigma
    Formula Description (Result)
    =NTNORMDIST((A2-A3)/A4,TRUE) Cumulative distribution function for the terms above
    =NTNORMDIST((A2-A3)/A4,FALSE) Probability density function for the terms above

    Sample distribution

  • Function reference : NTNORMDIST
  • NtRand Function NTNORMDIST is same as excel function NORMSDIST when 2nd. argument=TRUE.

Quantile

  • Inverse function of cumulative distribution function
    F^{-1}(P)=\sigma\Phi^{-1}(P)+m

    where \Phi(\cdot) is cumulative distribution function of standard normal distribution.

  • NORMSINV is an excel function
  • How to compute this on Excel.
     
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    A B
    Data Description
    0.7 Probability associated with the distribution
    1.7 Value of parameter M
    0.9 Value of parameter Sigma
    Formula Description (Result)
    =A4*NORMSINV(A2)+A3 Inverse of the cumulative distribution function for the terms above

Characteristics

Mean – Where is the “center” of the distribution? (Definition)

  • Mean of the distribution is given as m.

Standard Deviation – How wide does the distribution spread? (Definition)

  • Standard deviation of the distribution is given as \sigma.

Skewness – Which side is the distribution distorted into? (Definition)

  • Skewness of the distribution is given as 0.

Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

  • Kurtosis of the distribution is given as 0.

Random Numbers

x

  • How to generate random numbers on Excel.
     
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    Data Description
    0.5 Value of parameter M
    0.5 Value of parameter Sigma
    Formula Description (Result)
    =A3*NTRANDNORM(100)+A2 100 Normal deviates based on Mersenne-Twister algorithm for which the parameters above

    Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A5:A104 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions

  • Generating random numbers based on Mersenne Twister algorithm: NTRANDNORM

Reference

 


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