Discussion Overview
The discussion revolves around the conditions under which two marginally normal random variables can be considered jointly normally distributed. Participants explore whether the presence of marginal normality and normal conditional distributions guarantees a bivariate normal distribution or if other joint distributions are possible.
Discussion Character
- Debate/contested
- Technical explanation
- Mathematical reasoning
Main Points Raised
- Some participants assert that if two random variables are marginally normal and all conditional distributions are normal, this does not necessarily imply that the joint distribution is bivariate normal.
- Others argue that independence is not required for the joint distribution to be bivariate normal, but if independence exists, it guarantees that the joint distribution is bivariate normal.
- A participant suggests that even with normal marginal and conditional distributions, one can construct dependencies that violate the conditions for a bivariate normal distribution.
- Another participant emphasizes that the definition of a multivariate normal distribution includes the requirement that every linear combination of its components must also be normally distributed.
- Some participants propose that the additional condition of marginal normality alongside normal conditional distributions may restrict the joint distribution to be bivariate normal, but this remains contested.
- A later reply introduces a hypothetical scenario where conditional variances are inversely proportional to the other variable, suggesting that this could lead to a non-elliptical contour, thus challenging the bivariate normality assumption.
- One participant expresses uncertainty about the necessity of constant conditional variances as a missing condition for establishing joint normality.
- Another participant argues that correlation does not preclude normality and challenges the assertion that non-zero correlation leads to non-normal marginal distributions.
Areas of Agreement / Disagreement
Participants do not reach a consensus. There are multiple competing views regarding the implications of marginal and conditional normality on joint distribution, with ongoing debate about the necessity of independence and the role of conditional variances.
Contextual Notes
Some participants note that the definition of a multivariate normal distribution is complex and includes conditions beyond those discussed, such as the behavior of linear combinations of the variables. There is also mention of potential missing conditions that could affect the conclusions drawn.
