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# Intuition for why independence matters for variance of sum

AP.STATS:
VAR‑5 (EU)
,
VAR‑5.E (LO)
,
VAR‑5.E.2 (EK)
,
VAR‑5.E.3 (EK)

## Video transcript

so in previous videos we talked about the claim that if I have two random variables x and y that are independent then the variance of the sum of those two random variables or the difference of those two random variables is going to be equal to the sum of the variances so that if you have independent random variables your variation is going to increase when you take a sum or a difference and we've built a little bit of intuition there what I want to talk about in this video it's really about building even more intuition is get a gut feeling for why this independence is important for making this claim and to get that intuition let's look at two random variables that are definitely random variables but that are definitely not independent so let's say let's let X is equal to the number of hours that the next person you meet so I'll say random person random person slept yesterday and let's say that Y is equal to the number of hours that same person person was awake yesterday and appreciate why these are not independent random variables one of them is going to completely determine the other if I slept eight hours yesterday then I'm going then I would have been awake for 16 hours if I slept for 16 hours and I would have been awake for eight hours we know that X plus y even though they're random variables and there could be variation in X and there could be variation in Y but for any given person remember these are still based on that same person X plus y is always going to be equal to 24 hours so these are not independent not in the pendant if you're given one of the variables it would completely determine what the other variable is the probability of getting a certain value for one variable is going to be very different given what value you got for the other variable so they're not independent all so in this situation if someone said let's just say for the sake of argument that the variance of X the variance of X is equal to I don't know let's say it's equal to 4 and the units for variance it would be squared hours so 4 hours squared we could say that the standard deviation for X in this case would be 2 hours and let's say that the variance or let's say the standard deviation of Y is also equal to 2 hours and let's say that the variance of Y variance of Y well it would be the square of the standard deviation so it would be 4 hours 4 hours squared would be our units so if we just tried to blindly say oh I'm just gonna apply this this little expression this claim we had without thinking about the independence we would try to say well then the variance of X plus y the variance of X plus y must be equal to the sum of their variances so it would be 4 plus 4 so is it equal to 8 hours squared well that doesn't make any sense because we know that a random variable that is equal to X plus y that this is always going to be 24 hours in fact it's not going to have any variation X plus y is always going to be 24 hours so for these two random variables because they are so connected they are not depend independent at all this is actually going to be 0 there is zero variance here X plus y is always going to be 24 at least on earth where we have 24-hour day I guess if someone lived on another planet or something that it could be slightly different and we're assuming that we have an exactly 24-hour day on earth so this is to give you a gut sense of why independence matters for making this claim and if you have things that are not independent it gives you a good sense for why this claim doesn't hold up as much
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