Finding mean of Y if Y=(X1+X2+X3)/3 given mean and variance of x's

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Discussion Overview

The discussion revolves around calculating the mean and variance of a sample mean Y derived from three random variables X1, X2, and X3, each with a specified mean and variance. The participants explore whether the independence of the random variables affects the ability to compute these statistics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that the mean of Y can be computed as μ, while others express uncertainty about this conclusion due to the lack of information on the independence of the random variables.
  • One participant argues that averaging is linear and that dependence does not affect the expected value, asserting that E(Y) = μ.
  • There is a question regarding the calculation of the variance of Y, with some participants proposing that it could be 1/3 if independence is assumed.
  • Concerns are raised about the context of the problem and how it might influence the interpretation of the results.
  • One participant expresses a lack of resources or notes on combining distributions, indicating a need for further study on the topic.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the mean of Y can be definitively stated as μ without additional information about the independence of the random variables. There are competing views on the implications of independence for calculating variance.

Contextual Notes

Participants mention the importance of understanding the distributions of the random variables and the context of the problem, which remains unspecified. There is also a reference to the need for notes or resources on combining distributions, indicating potential gaps in knowledge.

Who May Find This Useful

This discussion may be useful for students or individuals studying statistics, particularly those interested in the properties of random variables and the implications of independence on statistical calculations.

ellyezr
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Let X1, X2, X3 be three random variables. Suppose all three have mean μ and variance 1. The sample mean is Y = (X1 +X2 +X3)/3.
(a) Can you compute the mean of Y? If so, what is it? If not, why not?

I have that it is either μ OR that it is not possible to find, since we don't know if they are independent or not (as it says later in the question). I have a strong feeling that it is the latter, but I am not sure.

(b) If we assume that the three random variables are independent, what would the variance of Y be?
1/3 right? Just to be sure.
 
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You should have notes on how to combine distributions.
How did you calculate the values you have suggested?
 
y=(x1+x2+x3)/3
x1 = x2 = x3 = mu
y=(mu+mu+mu)/3
y=3mu/3=mu

or can you not do that because you don't know if they are independent or not?

and no, I don't notes on that - trust me, I looked before I posted.
 
OK - well the first part seems to be saying you know nothing about the distributions on the ground that they are random. However, since the means are the same, does it make a difference?

The second part says they are independent - but nothing else - does that matter?

Or is the context of the problem important for figuring out what it all means?

Since you have no notes on this, you should try looking some up.
It would help me help you if I knew what level this should be answered at and if this forms part of a formal course.
 
ellyezr said:
Let X1, X2, X3 be three random variables. Suppose all three have mean μ and variance 1. The sample mean is Y = (X1 +X2 +X3)/3.
(a) Can you compute the mean of Y? If so, what is it? If not, why not?

I have that it is either μ OR that it is not possible to find, since we don't know if they are independent or not (as it says later in the question). I have a strong feeling that it is the latter, but I am not sure.

(b) If we assume that the three random variables are independent, what would the variance of Y be?
1/3 right? Just to be sure.

a) Averaging is linear - dpendence is irrelevant. E(Y) = μ
b) Yes.
 
thank you mathman
 

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