Further Maths question: Hard - stats

The discrete random variable R has the distribution Po(m).
Use an algebraic method to find the range of values of m for which the single most likely value of R is 7. [7 marks]

So, you need 7 to be the single mode of the distribution.

Given the shape of the Poisson distribution, you just need P(R=6)< P(R=7), and P(R=8) < P(R=7).

So, just work out the probabilities of each of those, and evaluate the inequalities to find the restrictions on m.