Question

x1, x2, and x3 all have a normal distribution with mean 1 and variance 1 and they are all independent of each other. what is the variance of (2*x1 x2 -x3)?

Answer 1

The variance of the combination of independent normally distributed variables (2*x1 + x2 - x3) is calculated by summing the variances of each, resulting in a combined variance of 6.

Step-by-step explanation:

To determine the variance of the combination of the independent normally distributed variables 2*x1, x2, and -x3, with each having a mean of 1 and a variance of 1, we use the properties of variance for independent random variables.

Since each variable is independent, the variance of the sum of these variables is the sum of their variances. The variance of a constant multiplied by a random variable is the square of the constant times the variance of the variable.

The variance for 2*x1 is (2^2)*Var(x1) = 4*1 = 4. The variance for x2 is simply Var(x2) = 1 since there is no constant multiplier. And, for -x3, the constant multiplier is -1, thus the variance is (-1^2)*Var(x3) = 1*1 = 1.

Now, combining these variances, we get Var(2*x1 + x2 - x3) = Var(2*x1) + Var(x2) + Var(-x3) = 4 + 1 + 1 = 6.

So, the variance of (2*x1 + x2 - x3) is 6.