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As an example, let's imagine that we have the following four variables measured in 500 people:

H - height in centimeters

W - weight in kilograms

B - whether the person is a member of a basketball team (binary variable)

S - whether the person is a member of a sumo wrestling club (binary variable)

Is there a joint distribution for H, W, B and S?

J. Doe
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  • Of course. But I would assume the distribution you mean a probability measure on $\mathbb{R}^4$ where the $(H,W,B,S)$ takes value. \begin{align} \mu(A_1\times A_2\times A_3\times A_4) = P(H\in A_1, W\in A_2,B\in A_3, S\in A_4) \end{align} where $A_i$ are 'measurable' sets in $\mathbb{R}$. However the joint distribution might not be the product probability measure. The joint probability measure I think should always exist – Zorualyh Feb 03 '23 at 14:56

1 Answers1

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Of course. The support of the joint distribution of $(X,Y)$ can be pretty much "any" subset of ${\mathbb R}^2$, for example, $\{0,1\}\times {\mathbb R}$, if which case $X$ is Bernouilli and $Y$ can be continuous.

van der Wolf
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  • But what would be the parameters of that distribution, given that the Bernoulli distribution and e.g. the normal distribution don't have the same parameters? – J. Doe Feb 03 '23 at 15:59
  • The parameters can be anything – van der Wolf Feb 04 '23 at 20:36
  • I'm sorry, that's not helpful – J. Doe Feb 05 '23 at 09:41
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    The parameter of Bernoulli distribution p can be any number between 0 and 1. The parameters of normal can be e.g. any positive numbers. I hope this is clearer. – van der Wolf Feb 06 '23 at 05:42
  • Unfortunately, it is not clearer. You are speaking now about the parameters of these univariate distributions, Bernoulli and normal. You are furthermore speaking not about the parameters themselves, but about which values they support. For example, the parameters of the normal distribution are called the mean and the standard deviation. My question is what would be the names of the parameters of this joint distribution of continuous and binary variables - they can't be the mean and the standard deviation because the binary variables in the joint distribution wouldn't support that – J. Doe Feb 06 '23 at 15:57
  • As far as I am aware, there are no specific "names" of such mixed distributions; you can find a list of common distributions in many books/internet, but they usually are either jointly continuous or jointly discrete. What I am saying is that you can have a joint distribution which does not belong to either of these two classes, it's just it wouldn't have an "established" name. – van der Wolf Feb 07 '23 at 10:05