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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

• 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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A B
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