The sequence of random variables $\{X_n\}_{n\in\mathbb N}$ is said to be weighted modulus $\alpha\beta$-statistically convergent in probability to a random variable $X$ [16] if for any $\varepsilon,\delta>0$, \[ im_{noıfty}\frac1{T_{lpha\beta(n)}}|\{keq T_{lpha\beta(n)}:t_khi(P(|X_k-X|\geqǎrepsilon))\geqẹlta\}| \] where $\phi$ be a modulus function and $\{t_n\}_{n\in\mathbb N}$ be a sequence of real numbers such that $\underset{n\to\infty}{\lim\inf}\,t_n>0$ and $T_{\alpha\beta(n)}=\sum_{k\in[\alpha_n,\beta_n]}t_k\forall n\in\mathbb N$. In this paper we study a related concept of convergence in which the value $\frac1{T_{\alpha\beta(n)}}$ is replaced by $\frac1{C_n}$, for some sequence of real numbers $\{C_n\}_{n\in\mathbb N}$ such that $C_n>0\forall n\in\mathbb N$, $\lim_{n\to\infty}C_n=\infty$ and $\underset{n\to\infty}{\lim\sup}\frac{C_n}{T_{\alpha\beta(n)}}<\infty$ (like [30]). The results are applied to build the probability distribution for quasi-weighted modulus $\alpha\beta$-statistical convergence in probability, quasi-weighted modulus $\alpha\beta$-strongly Cesàro convergence in probability, quasi-weighted modulus $S_{\alpha\beta}$-convergence in probability and quasi-weighted modulus $N_{\alpha\beta}$-convergence in probability. If $\{C_n\}_{n\in\mathbb N}$ satisfying the condition $\underset{n\to\infty}{\lim\inf}\frac{C_n}{T_{\alpha\beta(n)}}>0$, then quasi-weighted modulus $\alpha\beta$-statistical convergence in probability and weighted modulus $\alpha\beta$-statistical convergence in probability are equivalent except the condition $\underset{n\to\infty}{\lim\inf}\frac{C_n}{T_{\alpha\beta(n)}}=0$. So our main objective is to interpret the above exceptional condition and produce a relational behavior of above mention four convergences.