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Suppose you have IID standard normal variables $X_1 ,..., X_n$. Suppose each $X_i$ is paired with a real valued constant $C_i$. From this, we define new random variables, $Y_i$, such that if $X_i \leq C_i$, then $Y_i = 1$, and otherwise, $Y_i = 0$. How would you find a joint distribution for the $Y_i$?

Since the $X_i$ are IID, I guess that means the $Y_i$ are also IID. So my idea is I can just find the pdf of a single $Y_i$, and then the joint pdf will be the product of all these individual pdfs. I'm not sure how to find the distribution function for a single $Y_i$ though. I'm confused about how I can formulate a distribution function when its in terms of another random variable being above or below some constant.

Any suggestions appreciated.

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The $Y_i$ are each discrete (Bernoulli) random variables so they and the joint distribution does not have a density

Since the $c_i$ vary, the $Y_i$ are not identically distributed, though they are independent

But you can say $P(Y_i=1)=\Phi(c_i)$ and $P(Y_i=0)=1-\Phi(c_i)=\Phi(-c_i)$ using the cumulative distribution function of a standard normal, and then (using independence) multiply for the joint distribution.

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  • $\begingroup$ Apologies, I meant to just say "joint distribution" instead of "density". Does this change your answer? $\endgroup$
    – user790852
    Commented May 20, 2020 at 21:46
  • $\begingroup$ @IanVenter You would be multiplying the probability mass functions $\endgroup$
    – Henry
    Commented May 20, 2020 at 21:49

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