[M12] The uniform distribution over the countable sample space S = N can be defined as the probability density function L(x)=22l(x)1 or alternatively, 6 2(l(x)

[M12] The uniform distribution over the countable sample space S = N can be defined as the probability density function L(x)=2−2l(x)−1 or alternatively, 6

π2(l(x) + 1)2 2−l(x)

.

(a) Show that in both cases 

x∈S L(x) = 1.

(b) Let S1, S2,... be a sequence of sample spaces with Sn = {x : l(x) =

n}. Show that the probability density function Ln(x) = L(x|l(x) = n)

assigns probability Ln(x)=1/2n to all x of length n, and zero probability to other x’s, for n = 1, 2,....