  In this article, we address Amit--Ashurst conjecture on lower bounds of a
probability distribution associated to a word on a finite nilpotent group. We
obtain an extension of a result of Camina, Iniguez, and Thillaisundaram by
providing improved bounds for the case of finite nilpotent groups of class $2$
for words in an arbitrary number of variables, and also settle the conjecture
in some cases. We achieve this by obtaining words that are identically
distributed on a group to a given word. In doing so, we also obtain an
improvement of a result of Iniguez and Sangroniz using elementary techniques.
