sequence $b=\{b_k, k\ge 1\}$, let $\mathcal{P}_b(n)$ be the set of all weighted partitions of the positive integer $n$ and let $p_b(n)=\mid\mathcal{P}_b(n)\mid$ be its cardinality. It is known that the generating function $f_b(x)$ of the numbers $p_b(n)$ is of Euler’s type, namely, $$\label{euler}
f_b(x)=1+\sum_{n=1}^\infty p_b(n)x^n =\prod_{k=1}^\infty
(1-x^k)^{-b_k},\quad \mid x\mid<1$$ (see \[24; Sect. 3.14\]). We introduce the uniform probability measure $\mathbb{P}=\mathbb{P}_{n,b}$ on the set of weighted partitions of $n$ assuming that the probability $1/p_b(n)$ is assigned to each $n$-partition with weight sequence $b$. In this paper we focus on the size of the largest part $X_n$ of a random weighted partition of $n$. With respect to the probability measure $\mathbb{P}$, $X_n$ becomes a random variable, defined on the set $\mathcal{P}_b(n)$. It is also well-known that $$\label{df}
f_{m,b}(x)=1+\sum_{n=1}^\infty p_b(n)\mathbb{P}(X_n\le m)x^n
=\prod_{k=1}^m (1-x^k)^{-b_k},\quad m\ge 1$$ (see \[24; Sect. 3.15\]).

The asymptotic behavior of the combinatorial numbers $p_b(n)$ (the Taylor coefficients in (\[euler\])) will play important role in our further analysis. A fairly general scheme of assumptions on the parametric sequence $b$ was proposed by Meinardus \[13\] (see also \[2; Chap. 6\]), who found an asymptotic expansion for the numbers $p_b(n)$ as $n\to\infty$. The same asymptotic was also obtained in \[10\], where one of the 