Let $X|\theta \sim F$ with $\theta$ a random vector and $X$ a random variable. Is it true that $$ E[X] = E[E[X|\theta]] $$ even though $\theta$ is a vector? If so, can one say that if $X|\mu,\sigma \sim N(\mu, \sigma^2)$ then $E[X] = E[E[X|\mu,\sigma]] = E[\mu]$?
2026-03-25 20:41:35.1774471295
Is the tower property true conditional on vector?
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As @geetha290krm mentioned, it is true.