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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?

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