# Derive the CDF of the sum of two independent random variables [duplicate]

Some notational remarks before presenting the question:

• $$k<\infty$$, with $$k\in \mathbb{N}$$.

• $$\lambda\equiv (\lambda_1,...,\lambda_k)$$, $$\lambda_j\in [0,1]^k$$ $$\forall j$$, and $$\sum_{j=1}^k\lambda_j=1$$.

• $$\mu\equiv (\mu_1,...,\mu_k)$$, $$\mu_1<\mu_2<...<\mu_k$$, and $$\mu_j\in \mathbb{R}$$ $$\forall j$$.

• $$\bar{\mathbb{R}}$$ denotes the extended real line $$\mathbb{R}$$.

• $$G:\bar{\mathbb{R}}\rightarrow [0,1]$$ denotes a CDF with associated probability distribution that is symmetric around zero.

• "$$\sim$$" denotes "distributed as". "$$\perp$$" denotes "stochastically independent".

• Consider the CDF $$\Delta_{k,\lambda,\mu}: \bar{\mathbb{R}}\rightarrow [0,1]$$ prescribed by $$\Delta_{k,\lambda, \mu}(x)= \sum_{j=1}^k \lambda_j 1\{x\geq \mu_j\} \text{ }\forall x \in \bar{\mathbb{R}}$$

• Consider the CDF $$F_{\lambda, \mu,G}: \bar{\mathbb{R}}\rightarrow [0,1]$$ prescribed by $$F_{\lambda, \mu,G}(x)=\sum_{j=1}^k \lambda_j G(x-\mu_j) \text{ }\forall x \in \bar{\mathbb{R}}$$

The question:

Take the random variables $$Z\sim G$$, $$Y\sim \Delta_{k,\lambda, \mu}$$, and $$Z\perp Y$$. Then $$Y+Z\sim F_{k,\lambda,\mu}$$.

Could you help me to formally show this claim?