Let X be a random variable with probability density function given by where and are strictly positive constants Such a random variable is said to have a Weibull distribution with shape parameter 0 and scale parameter 0, and we denote this fact by X W(, ) 1 Compute the cdf FX of X 2 The logarithm of a Weibull random variable has a distribution known as the Gumbel distribution (with the same parameters) Describe (in words) an algorithm to generate a random variable having such a Gumbel distribution

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Question: Let X be a random variable with probability density function given by where and are strictly positive constants. Such a random variable is

Let X be a random variable with probability density function given by where α and β are strictly positive constants. Such a random variable is said to have a Weibull distribution with shape parameter α > 0 and scale parameter β > 0, and we denote this fact by X ∼ W(α, β).

1. Compute the cdf. FX of X.

2. The logarithm of a Weibull random variable has a distribution known as the Gumbel distribution (with the same parameters). Describe (in words) an algorithm to generate a random variable having such a Gumbel distribution.

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