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# Variance of sum and difference of random variables

Intuition for why the variance of both the sum and difference of two independent random variables is equal to the sum of their variances.

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• why we can't take the sum of standard deviation? why we can only take the sum of variance? • If X and Y are independent, then Var(X + Y) = Var(X) + Var(Y) and Var(X - Y) = Var(X) + Var(Y).

However, this does not imply that the same is true for standard deviation, because in general the square root of the sum of the squares of two numbers is usually not the sum of the two numbers. Note that

SD(X + Y) = sqrt[Var(X) + Var(Y)] = sqrt{[SD(X)]^2 + [SD(Y)]^2}, which is not SD(X) + SD(Y).
SD(X - Y) = sqrt[Var(X) + Var(Y)] = sqrt{[SD(X)]^2 + [SD(Y)]^2}, which is not SD(X) + SD(Y).

Have a blessed, wonderful day!
• In this video I couldn't get my head around what X-Y looks like in real life. X+Y is easy: its just pouring a bowl of cereal - but what does X-Y mean? • Suppose that X is a random variable that represents how many times a person scratches their head in a 24 hours period and Y is a random variable that represents the number of times a person scratches their nose in the same time period.
X+Y represents the sum, meaning how many times they scratch their head and nose combined.
X-Y represents the difference, meaning how many more times they scratch their head compared to their nose.
• I don't get that both Var(X + Y) and Var (X - Y) equal Var(X) + Var (Y). In what way?
Thank you. • At :
E(X+Y) = 20 oz
Var(X+Y) = 1 but........ 18 <= X+Y <= 22 that means the actual variance is 2 ( 2oz above the mean "20oz" and 2oz below the mean) and not 1. How come? • What if it's not independent? How do we derive the E(X+Y) and VAR(X+Y)? • Interesting question! If we don’t assume that X and Y are independent, it is still always true that E(X+Y) = E(X)+E(Y). However, it is not necessarily true that Var(X+Y) = Var(X)+Var(Y). Instead, the more general rule Var(X+Y) = Var(X)+2Cov(X,Y)+Var(Y) is always true, where Cov(X,Y) is the covariance of X and Y, which is defined as E(XY)-E(X)E(Y).

Have a blessed, wonderful day!
• If the variance of both X+Y and X-Y is Var(X)+Var(Y), what happens if you keep combining and separating the cereal box and the bowl of cereal? Intuitively the variance won't go to infinity but according to the video it would? Or will they not be independent anymore once you combine the box of cereal with the bowl of cereal? • yes it goes to infinity (only if you keep doing this infinitely for sure)

1. you can think of a variance as an error from the "true" value of an object being measured
var(X+Y) = an error from measuring X, measuring Y, then adding them up
var(X-Y) = an error from measuring X, measuring Y, then subtracting Y from X

2. if you do this once, you may have var(X+Y)+var(X-Y) as your total error and they are not cancelling out one another, probably unlike your intuition (there's no such thing as a negative error or variance)

3. if you do this infinitely many times, your total errors must climb to infinity. (unless you had the golden hand of making no error at all) though you keep having just one box of cereal, some of which would be in the bowl depending on the timing of your halting this infinite experiment

variance doesn't tell you the "true" value of something, but how far your measurement is from it. so the more measurements, the larger the error
(1 vote)
• Sal here tried to give the intuition of why we add the variance in both cases. Didn't he use range to explain it. As 15<=X=<17 represents the range not the variance. Isn't variance the square of the difference between observed values and mean, and how can it be taken in a similar sense to range?
Is there any other way to get intuition about adding up variance? • if you push Z (a value representing a standard deviation, and thus a variance of a measurement) to the small and large enough values (say -4 and 4), you can be certain that there are no measurable values below and above this spectrum

and you can call it a range

in other words, a range is an extreme version of variance. and it's a lot simpler to do the math with a range than variable in many cases. that's why Sal used a range to explain this concept on variable a bit more simply, i believe
(1 vote)
E.g.
Cereal was 15 < x 17
Bowl was 14 < x < 16 • If Var(X+Y)= Var(X)+ Var(Y)
And Variance=(Standard Deviation Square)^2
Then why can't we take the standard deviations of both random variables and calculate their squares and add them up? That would give us the variance, right?

Let's take the example shown in the video
σ(X)= 0.8 oz
Var(X)= (σ(X))²
∴ Var(X)= (0.8)²=0.64

σ(Y)= 0.6 oz
Var(Y)= (σ(Y))²
∴ Var(Y)= (0.6)²=0.36

Var(X+Y)= Var(X)+Var(Y)=0.64+0.36
∴Var(X+Y)=1

We can also calculate the standard deviation of X+Y using this variance-
Variance= σ²
σ= √variance
σ(X+Y)= √var(X+Y)= √1
∴σ(X+Y)= 1

Why isn't the variance calculated this way?

🎊Happy New Year🎊
Thank you
(1 vote) • Why is the variance equal to the standard deviation squared?
(1 vote) 