Calculating Variance: Add X & Y To Get 32*Var[X] + Var[Y]?

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waealu
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Hi, I have a quick question about variance.

If you are trying to compute Var[3X-Y] where Var[X]=1 and Var[Y]=2,
the correct formula is 32*Var[X] + Var[Y],
but why do you add the variances of x and y?

If you were computing Var[3X+Y] would it also be 32*Var[X] + Var[Y] ?

Thanks, Erik
 
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  • #2
waealu said:
Hi, I have a quick question about variance.

If you are trying to compute Var[3X-Y] where Var[X]=1 and Var[Y]=2,
the correct formula is 32*Var[X] + Var[Y],
but why do you add the variances of x and y?
Your textbook probably has a theorem about Var[X + Y] and Var[kX]
waealu said:
If you were computing Var[3X+Y] would it also be 32*Var[X] + Var[Y] ?
Yes.
 
  • #3
Hey waealu.

For the proof, the easiest way is to use VAR[X] = E[X^2] - {E[X]}^2 and then prove the identity. You do it for the continuous and discrete cases and you'll get the same result for both.

Intuitively though, the variance is a squared positive quantity which means that the variance has to always increase (and recall that variances are always positive for a random variable).
 
  • #4
waealu said:
Hi, I have a quick question about variance.

If you are trying to compute Var[3X-Y] where Var[X]=1 and Var[Y]=2,
the correct formula is 32*Var[X] + Var[Y],
but why do you add the variances of x and y?

If you were computing Var[3X+Y] would it also be 32*Var[X] + Var[Y] ?

Thanks, Erik

These are only true if X and Y are independent (or, at least, uncorrelated) random variables. For independent (or uncorrelated) X and Y and constants a, b we have
[itex] \text{Var}[aX + bY] =a^2 \text{Var}(X) + b^2 \text{Var}(Y).[/itex]

RGV
 

1. What is the formula for calculating variance?

The formula for calculating variance is to take the sum of squared differences between each data point and the mean, divided by the total number of data points.

2. What is the significance of adding X and Y in the formula?

Adding X and Y in the formula allows for calculating the variance of two variables together, taking into account their individual variances as well as their covariance.

3. Why is it important to multiply Var[X] by 32?

Multiplying Var[X] by 32 is a mathematical adjustment that is commonly used to scale the variance for a larger sample size.

4. What does Var[Y] represent in the formula?

Var[Y] represents the variance of the second variable, Y, in the calculation of the combined variance.

5. Can this formula be applied to non-numerical data?

No, this formula is specific to calculating the variance of numerical data and cannot be applied to non-numerical data.

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