Let the variable be

Yc = theta_1 + theta_1*theta_2+ theta_1*theta_2 theta_3 + ....

where theta_i are i.i.d uniform random variables (i = 1,2,...) having

CDF such that F(x) = 0, x <=0. F(x) = x,

F(x) = 0<x<c and

F(x) = 1 , for x >=c.

Question,

1)for what values of c > 0 is E(Yc) finite.

2)Also, for what value of c > 0, Yc (sum converges with probability 1).