# Determine Weibull parameters (scale and shape) from hazard rate

I have a hazard rate given by a 2-parameters Weibull distribution, in the form:

$$h(t) = \cfrac{B}{A} \, \left({\cfrac{t}{A}}\right)^{B - 1}$$

where $A=$ scale parameter and $B=$ shape parameter. I don't have a lot of knowledge about probability distributions, but I would like to use this one to reproduce the famous bathtub curve.

How can I establish the proper value of $A$ and $B$ in order to obtain 3 distributions like these:

For the green one (constant hazard rate) I know that the ordinate of the line is given by $B/A$, therefore in the example it is 0.1 ($B/A=1/10=0.1$); but for the other two I would like to establish some formula that permits me to establish the value of A and B in order to:

1. do not exceed never the value of $h(t) = 1$
2. decrease (or increase) asymptotically until to (from to) the same value of the constant part (e.g. 0.1 in the previous example) so I can "stick" them together at the end like in the following image.

• The hazard of a Weibull distribution is always monotonic - increasing, decreasing or staying constant, but not first decreasing and then increasing. So there is no way to "reproduce the famous bathtub curve" for h(t) using a Weibull hazard. Your questions are not clear to me. Your graphs seem to indicate the parameters A and B used to produce them and you have also realised that the intercept will be B/A. You seem to want to have multiple hazard functions to "'stick' them together", but your goal is not clear to me. – Rob Hall Jul 15 '15 at 12:38
• Hi Rob, thanks for your answer. My goal is to find a relationship that permits me to establish where my curve will start and end depending on the scale and shape that I choose. In the case of the green distribution it was just B/A because it is a constant line. But what in the case of B<1 and B>1? In few words, if I want that my distribution to start from h(t1) = n1 and ends at h(t) = n2, regardless of the shape that it will assume, how can I find a formula that relates n1 and n2 to A and B? (e.g. above end of red line and beginning of yellow are 0.1) – Ggiioo Jul 15 '15 at 12:53