# Maximize or find an upper bound of the function $kx^{k-1}\exp(-\mu(x^k-x))$

I was programming some random variable simulation using the acceptance-rejection method and I encounter with the Weibull$(k,\lambda)$ distribution. This random variable is posible to simulate with this method when $k \geq 1$. To do this I use a exponential random variable of parameter $\lambda^{-k}$ but I need to find a upper bound of the function $$h(x)=kx^{k-1}\exp(-\mu(x^k-x)) \quad \text{with} \mu=\lambda^{-k}$$ Because if $f_{k,\lambda}$ is the probability density of a Weibull$(k,\lambda)$ random variable and $g_{\lambda^{-k}}$ is the probability density of a Exp$(\lambda^{-k})$ random variable then $$f_{k,\lambda} \leq C g_{\lambda^{-k}}$$ with $C= \max_{x > 0} h(x)$

I try to use good old derivation to find the maximum but the equation that came up is $$\lambda^{-k}(x-kx^k)=1-k$$ which I can't solve.

Any help finding a upper bound will be appreciated!

• I do not think that you could find any closed form solution. Probably, numerical methods would be the only way. What are the ranges (or values) of $k$ and $\lambda$ ? – Claude Leibovici Aug 16 '14 at 4:29