Consider a random variable x with the following values and the corresponding probabilities: {x = 1, p(x

Consider a random variable ∆x with the following values and the corresponding probabilities:
{Δx = 1, p(Δx = 1) = .3},
{Δx = –0.5, p(Δx = –0.5) = .2},
{Δx = .2, p(Δx = .2) = .5}.
(a) Calculate the mean and the variance of this random variable.
(b) Change the mean of this random variable to .05 by subtracting an appropriate constant from ∆x. That is, calculate
∆y = ∆x – μ
Such that ∆y has mean .05.
(c) Has the variance changed?
(d) Now do the same transformation using a change in probabilities, so that again the variance remains constant.
(e) Have the values of ∆x changed?