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# Triangular distribution

## Shape of Distribution

### Basic Properties

• Three parameters $a, b, c$ are required (How can you get these).
$a

These parameters are minimum value of variable, maximum value of variable and mode of the distribution respectively.

• Continuous distribution defined on bounded range $a\leq x \leq b$
• This distribution can be symmetric or asymmetric.

### Probability

• Cumulative distribution function
$F(x)=\begin{cases}\frac{(x-a)^2}{(b-a)(c-a)}\quad&(a\leq x
• Probability density function
$f(x)=\begin{cases}\frac{2(x-a)}{(b-a)(c-a)}\quad&(a\leq x
• How to compute these on Excel.

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A B
Data Description
1.5 Value for which you want the distribution
1 Value of parameter Min
3 Value of parameter Max
1.4 Value of parameter Mode
Formula Description (Result)
=NTTRIANGULARDIST(A2,A3,A4,A5,TRUE) Cumulative distribution function for the terms above
=NTTRIANGULARDIST(A2,A3,A4,A5,FALSE) Probability density function for the terms above
• Function reference : NTTRIANGULARDIST

### Quantile

• Inverse function of cumulative distribution function
$F^{-1}(P)=\begin{cases}\sqrt{P(c-a)(b-a)}+a\quad&\left(P< \frac{c-a}{b-a}\right)\\-\sqrt{(1-P)(b-c)(b-a)}+b\quad&\left(P\geq \frac{c-a}{b-a}\right)\end{cases}[/latex]
• How to compute this on Excel.
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A B
Data Description
0.5 Probability associated with the distribution
1 Value of parameter Min
3 Value of parameter Max
1.4 Value of parameter Mode
Formula Description (Result)
=NTTRIANGULARINV(A2,A3,A4,A5) Inverse of the cumulative distribution function for the terms above
• Function reference : NTTRIANGULARINV
• ## Characteristics

### Mean - Where is the center'' of the distribution? (Definition)

• Mean of the distribution is given as
$\frac{a+b+c}{3}$ • How to compute this on Excel 1 2 3 4 5 6 A B Data Description 1 Value of parameter Min 3 Value of parameter Max 1.4 Value of parameter Mode Formula Description (Result) =NTTRIANGULARMEAN(A2,A3,A4) Mean of the distribution for the terms above • Function reference : NTTRIANGULARMEAN ### Standard Deviation – How wide does the distribution spread? (Definition) • Variance of the distribution is given as $\frac{a^2+b^2+c^2-ab-bc-ca}{18}$ Standard Deviation is a positive square root of Variance. • How to compute this on Excel 1 2 3 4 5 6 A B Data Description 1 Value of parameter Min 3 Value of parameter Max 1.4 Value of parameter Mode Formula Description (Result) =NTTRIANGULARSTDEV(A2,A3,A4) Standard deviation of the distribution for the terms above • Function reference : NTTRIANGULARSTDEV ### Skewness – Which side is the distribution distorted into? (Definition) • Skewness of the distribution is given as $\frac{\sqrt{2}(a+b-2c)(2a-b-c)(a-2b+c)}{5(a^2+b^2+c^2-ab-bc-ca)^{3/2}}$ • How to compute this on Excel 1 2 3 4 5 6 A B Data Description 1 Value of parameter Min 3 Value of parameter Max 1.4 Value of parameter Mode Formula Description (Result) =NTTRIANGULARSKEW(A2,A3,A4) Skewness of the distribution for the terms above • Function reference : NTTRIANGULARSKEW ### Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition) • Kurtosis is $-0.6$ . ## Random Numbers • Random number x is generated by inverse function method, which is for uniform random U, x=\begin{cases}\sqrt{U(c-a)(b-a)}+a\quad&\left(U< \frac{c-a}{b-a}\right)\\-\sqrt{(1-U)(b-c)(b-a)}+b\quad&\left(U\geq \frac{c-a}{b-a}\right)\end{cases}$
• How to generate random numbers on Excel.

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A B
Data Description
0 Value of parameter A
3 Value of parameter B
1.8 Value of parameter C
Formula Description (Result)
=NTRANDTRIANGULAR(100,A2,A3,A5,0) 100 triangular 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 A6:A105 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.