# Exponential Distribution Function and Derivation of the Equation

Let F(t) = 1- e ^(- lamda*t)

a) Show how to generate a random variable from the exponential distribution function shown above. Show derivation of the equation.

b) Generate 10000 random variables X and Y with cumulative distribution F(t); do this twice using lambda =2,3 (so 2 columns of 10000 numbers each).

c) Using the equation from a) and the data from b) construct a graph(s) that confirms lambda is the values used (ie 2 and 3)

Use Excel Data analysis and charts for the solution.

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#### Solution Preview

See the attached files.

a) It is given that, the cumulative distribution function is

F(t) = 1 - e^lambda*t

We know that the distribution of the distribution function F(t) is uniform, U[0,1].

Let u be a random umber from the Uniform distribution on [0, 1].

Now, we can generate a random number from F(t) = 1 - e^ -lambda*t using the equation

u = 1 - e^-lambda*t

That is,

e^-lambda*t = 1 - u

Taking natural logarithm on both sides we get,

-lambda*t = ln(1-u)

Therefore, t = (-1/lambda)ln(1-u).

Thus, the simulation procedure is first generate uniform random numbers u_1, and u_2 and compute

X = -1/2 ln(1-u_1) and Y = -1/3 ln(1-u_2). ...

#### Solution Summary

Exponential distribution functions and derivation of the equations are examined. The expert uses Excel Data Analysis and charts.