(A universal random number generator.)Let X have a continuous, strictly increasing cdf F. Let Y =...

(A universal random number generator.)Let X have a continuous, strictly increasing cdf F. Let Y = F(X). Find the density of Y. This is called the probability integral transform. Now let U ∼ Uniform(0,1) and let X = F−1(U). Show that X ∼ F. Now write a program that takes Uniform (0,1) random variables and generates random variables from an Exponential (β) distribution

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Probability Integral Transform method:

If has distribution, then such that or is an observation from the probability distribution , this means that we can generate observations from the distribution by generating random variables (which most software programs can do easily) and applying the transformation.

Suppose you want to generate instances of an exponential() random variable. The cdf is

The following R code generates 10 exponential random variables taking Uniform (0,1) random variables.

n_random = 10
beta = 2
U = runif(n_random, min = 0, max = 1)
Y = −(1/beta)*log(1-U)
Y

orchestra answered 1 year ago

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