Discussion Overview
The discussion revolves around finding the expected value of a geometric random variable X given the sum of two independent geometric random variables X and Y, denoted as Z=X+Y. Participants explore the conditional expectation E[X|Z=z] and its relationship with E[Y|Z=z], including various approaches to derive these values.
Discussion Character
- Mathematical reasoning
- Exploratory
- Debate/contested
Main Points Raised
- One participant calculates the conditional probability P(X=x|Z=z) and suggests that it equals 1/(z-1), questioning how to find the expected value from this.
- Another participant poses a trick question regarding the relationship between E[X|Z=z] and E[Y|Z=z], asking which is larger.
- A participant proposes that E[X|Z=z] + E[Y|Z=z] equals z, based on the expectation of Z given Z=z.
- Some participants discuss the possibility of using the uniform distribution to find the expected value, given the derived conditional probability.
- One participant attempts to derive E[X|Z=z] through summation and presents a calculation leading to z/2, questioning its validity.
- Another participant confirms the validity of the calculation and mentions an alternative approach referenced in a previous post.
- There is a repeated inquiry about the comparative sizes of E[X|Z=z] and E[Y|Z=z], with no clear resolution on the trick question posed.
Areas of Agreement / Disagreement
Participants express differing views on the relationship between E[X|Z=z] and E[Y|Z=z], and there is no consensus on the trick question regarding which expected value is larger. The validity of the calculations presented is also debated, with some participants affirming certain approaches while others remain uncertain.
Contextual Notes
Participants note that the calculations depend on the assumption of the uniform distribution for P(X=x|Z=z) and the limits of summation from x=1 to x=z-1. There are unresolved questions about the implications of these assumptions on the expected values derived.