Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable 2X.
A)μ = 80,σ = 5
B)μ = 42,σ = 5
C)μ = 42,σ = 10
D)μ = 80,σ = 10
E)μ = 80,σ = 20

Question

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable 2X. 
A)μ = 80,σ = 5 
B)μ = 42,σ = 5 
C)μ = 42,σ = 10 
D)μ = 80,σ = 10 
E)μ = 80,σ = 20

Quizplus · Accepted Answer

By definition, the mean of 2X is:

E[2X] = 2E[X]

If X has mean μ_X and standard deviation σ_X, then we have:

E[2X] = 2E[X] = 2μ_X

The variance of 2X is:

Var(2X) = Var(X + X) = Var(X) + Var(X) = 2Var(X)

Therefore, the standard deviation of 2X is:

σ_{2X} = sqrt(Var(2X)) = sqrt(2Var(X))

Using the given information, we have:

μ_X = 40
σ_X = 2.5

So:

E[2X] = 2μ_X = 80

σ_{2X} = sqrt(2Var(X)) = sqrt(2σ_X^2) = sqrt(2*(2.5)^2) = 5*sqrt(2)

Therefore, the answer is D) μ = 80, σ = 10.

Given Independent Random Variables with Means and Standard Deviations as Shown,find