Years ago Zeev Rudnick defined the
λ
\lambda
-Poisson generic sequences as the infinite sequences of symbols in a finite alphabet where the number of occurrences of long words in the initial segments follow the Poisson distribution with parameter
λ
\lambda
. Although almost all sequences, with respect to the uniform measure, are Poisson generic, no explicit instance has yet been given. In this note we give a construction of an explicit
λ
\lambda
-Poisson generic sequence over any alphabet and any positive
λ
\lambda
, except for the case of the two-symbol alphabet, in which it is required that
λ
\lambda
be less than or equal to the natural logarithm of
2
2
. Since
λ
\lambda
-Poisson genericity implies Borel normality, the constructed sequences are Borel normal. The same construction provides explicit instances of Borel normal sequences that are not
λ
\lambda
-Poisson generic.