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

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The mean of Y, defined as Y = (X1 + X2 + X3)/3, is μ, regardless of whether the random variables X1, X2, and X3 are independent or not, due to the linearity of expectation. For part (b), if the three random variables are assumed to be independent, the variance of Y is calculated as 1/3. The discussion emphasizes that independence does not affect the calculation of the mean but is relevant for variance. Understanding the context and level of the problem is important for accurate interpretation and calculation.
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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
 
I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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