2 Tried to provide a lead in answering the question.
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The sum of independent normally distributed variables will itself be normally distributed. This then implies that the probability of their sum being any particular value $Z$ is 0.

You might want to rethink your question in terms of putting an interval on Z - i.e., what is the probability that the result is within $[Z-\omega;Z+\omega]$.


Let $X_i \sim N(\mu_i,\sigma_i)$ and $X_j \sim N(\mu_j,\sigma_j)$ be independent normal random variables. Then their sum is given by $$X_{ij}=X_i+X_j \rightarrow X_{ij} \sim N(\mu_i+\mu_j,\sigma_i+\sigma_j)$$.

Given this, finding the probability that $X_{ij}$ is within some interval $[Z-\omega;Z+\omega]$ can be found by subtracting the CDF of $X_{ij}$ at $Z-\omega$ from the CDF at $Z+\omega$.

The sum of independent normally distributed variables will itself be normally distributed. This then implies that the probability of their sum being any particular value $Z$ is 0.

You might want to rethink your question in terms of putting an interval on Z - i.e., what is the probability that the result is within $[Z-\omega;Z+\omega]$.

The sum of independent normally distributed variables will itself be normally distributed. This then implies that the probability of their sum being any particular value $Z$ is 0.

You might want to rethink your question in terms of putting an interval on Z - i.e., what is the probability that the result is within $[Z-\omega;Z+\omega]$.


Let $X_i \sim N(\mu_i,\sigma_i)$ and $X_j \sim N(\mu_j,\sigma_j)$ be independent normal random variables. Then their sum is given by $$X_{ij}=X_i+X_j \rightarrow X_{ij} \sim N(\mu_i+\mu_j,\sigma_i+\sigma_j)$$.

Given this, finding the probability that $X_{ij}$ is within some interval $[Z-\omega;Z+\omega]$ can be found by subtracting the CDF of $X_{ij}$ at $Z-\omega$ from the CDF at $Z+\omega$.

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The sum of independent normally distributed variables will itself be normally distributed. This then implies that the probability of their sum being any particular value $Z$ is 0.

You might want to rethink your question in terms of putting an interval on Z - i.e., what is the probability that the result is within $[Z-\omega;Z+\omega]$.