# Homework help: how to find the following joint distribution?

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.

## 1 Answer

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.

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