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A random variable X has a mean of 10 and a standard deviation of 3. If each value of X is multiplied by 2, what will the new mean and standard deviation be?

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Finding the new mean is simple: If
E[X]=10, then
E[2X]=2E[X]=20.

Meanwhile, the standard deviation is the square root of the variance, which is given by


V[X]=E[X^2]-E[X]^2\implies V[2X]=E[(2X)^2]-E[2X]^2=4E[X^2]-(2E[X])^2


\implies V[2X]=4\underbrace{\left(E[X^2]-E[X]^2\right)}_(V[X])=12

and so the new standard deviation would be
√(V[2X])=√(12)=2\sqrt3.

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