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Statistics: Linear combination of normal variables?
The lengths of red pencils are normally distributed with mean 6.5 and standard deviation 0.23. Two red pencils are chosen at random. Calculate the mean and variance of their total lengths.
Ok, no probs for E(X) = mean
E(2X) = 2E(X) = 2(6.5) = 13 (no probs here)
But as I understand, var(kx) = k².var(x), so:
Var(2X) = (2²).Var(X) = (4)(0.23)² = 0.2116
But in answer sheet it says Var(X) = (2)(0.23)² = 0.1058
Can anyone clarify what's going on here? Am I wrong? Help appreciated.
1 Answer
- 1 decade agoFavorite Answer
You are mistaking the variance of 2 times a random variable with the variance of the sum of two independent random variables. If two variables X and Y are independent (or even uncorrelated) then Var[X+Y]= Var[X]+ Var[Y]
Now each pencil has length that is a random variable following the given normal distribution, they are not the same random variable! They just happen to have the same variance individually...
Thus you need Var[X+Y]= Var[X]+Var[Y] = .23^2 + .23^2