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It seems you assume $X$ is independent of $W_i$, and $W_i$ are jointly normal. Following this fact, $Z_i$ become jointly normal, and you can use the conditional distribution formula to find the joint PDF of $Z_i$. A similar approach for finding the joint PDF would be directly calculating the mean and covariance vector for $[Z_1, Z_2]$ ($Z$ is jointly normal because it's a linear transform of the random vector $[X,W_1,W_2]$). However, multivariate normal CDF has no closed form.

It seems you assume $X$ is independent of $W_i$, and $W_i$ are jointly normal. Following this fact, $Z_i$ become jointly normal, and you can use the conditional distribution formula to find the joint PDF of $Z_i$. A similar approach for finding the joint PDF would be directly calculating the mean and covariance vector for $[Z_1, Z_2]$. However, multivariate normal CDF has no closed form.

It seems you assume $X$ is independent of $W_i$, and $W_i$ are jointly normal. Following this fact, $Z_i$ become jointly normal, and you can use the conditional distribution formula to find the joint PDF of $Z_i$. A similar approach for finding the joint PDF would be directly calculating the mean and covariance vector for $[Z_1, Z_2]$ ($Z$ is jointly normal because it's a linear transform of the random vector $[X,W_1,W_2]$). However, multivariate normal CDF has no closed form.

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gunes
gunes
  • 58.2k
  • 4
  • 50
  • 88

It seems you assume $X$ is independent of $W_i$, and $W_i$ are jointly normal. Following this fact, $Z_i$ become jointly normal, and you can use the conditional distribution formula to find the joint PDF of $Z_i$. A similar approach for finding the joint PDF would be directly calculating the mean and covariance vector for $[Z_1, Z_2]$. However, multivariate normal CDF has no closed form.