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Intuition for why independence matters for variance of sum

Intuition for why independence matters for variance of sum.

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• Would someone put into words what is being measured by Var(X + Y)? I understand that of all the people in our sample, for both random variables, there was an average spread from the mean of 2 hours. What does it mean when we add these?
• idk may be because that is how it was supposed to be.
(1 vote)
• I want to ask 2 things here.
If the variances of dependent variables is 0 because they dependent on each other, like if this one changes the other one will be changed too thus there will no change in variances, Am I understand it correct?
and What about the mean of dependent variables?
(1 vote)
• In this particular case 𝑋 + 𝑌 is a constant, which is why Var(𝑋 + 𝑌) = 0.

This isn't always the case, though, and besides it's not very relevant.
What Sal wanted to show is that the equation
Var(𝑋 ± 𝑌) = Var(𝑋) + Var(𝑌) doesn't necessarily hold up if 𝑋 and 𝑌 are dependent.

– – –

For your second question, since the outcome of 𝑌 depends on the outcome of 𝑋, then the mean of 𝑌 depends on the mean of 𝑋.

In this case 𝜇(𝑋) is the number of hours that the average person slept yesterday, while 𝜇(𝑌) is the number of hours the average person was awake yesterday.
That gives us 𝜇(𝑌) = 24 − 𝜇(𝑋)