Best solutions to - "Illustrate the inverse transformation method for obtaining a..."

Ann Tice

Answered question

2021-11-25

Illustrate the inverse transformation method for obtaining a random observation from a given probability distribution?

Lounctirough

Beginner2021-11-26Added 14 answers

Step 1
The inverse transform method can be used to generate random numbers from various probability distributions.
For example,

X \sim E x p (λ)

So, the distribution function of X is:

F (x) = P (X \leq x)

= 1 - e^{- λ x}

To generate random numbers, first generate random numbers for a variable U where

U \sim U (0, 1)

such that:

U = F (x)

Step 2
Using Excel command RAND() random numbers for Uniform(0,1) can be generated.
Now, the formula for x is:

u = F (x) = 1 - e^{- λ x}

\Rightarrow \log (1 - u) = - λ x

\Rightarrow x = \frac{- 1}{λ} \log (1 - u)

Substituting the values of u generated on excel, we get the corresponding random numbers for x as below:
(Taking the value of

λ = 0.3

for example)

\begin{array}{cc} u & x = - \ln (1 - u) / 0.3 \\ 0.106619 & 0.3758067 \\ 0.546745 & 2.637670251 \\ 0.8496 & 6.314863841 \\ 0.238444 & 0.907970784 \\ 0.126093 & 0.449271194 \\ 0.118761 & 0.419152116 \\ 0.253265 & 0.97348524 \\ 0.231665 & 0.878432301 \\ 0.096053 & 0.336614596 \end{array}

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