On a Problem in Probability Theory

For continuous random variables, we study a problem similar to that considered earlier by one of the authors for discrete random variables. Let numbers
$$ N>0,\qquad E>0,\qquad 0\le\lambda_1\le\lambda_2\le\dotsb\le\lambda_s $$
be given. Consider a random vector $x=(x_1,\dots,x_s)$, uniformly distributed on the set
$$ x_j\ge0,\quad j=1,\dots,s;\qquad \sum_{j=1}^sx_j=N,\quad \sum_{j=1}^s\lambda_jx_j\le E. $$
We study the weak limit of $x$ as $s\to\infty$.