simple calculation question in statistic

Question

i've been solving questions for hours as a preparation, and my mind doesn't work. i've done most of the steps needed to finish the calculation, but i always think i'm missing something in it.

the question:

having $Y=\frac{x-t}{2}$, where x is a random variable(normal) with mean t and variance 4. i need to calculate E[(x-t)^2].

what i did:

so i obtained that Y as given is standard normal with deviation 1 and mean 0. i calculated $E[Y^2]=1$ because of mean 0 and variance 1, and i am stuck here. please help me.

thank you very much for that

Answer 1

You do not need any $Y$.

• $E[(X-t)^2]$ is by definition the variance of $X$, as $t$ is the mean of $X$.
• The variance of $X$ is also given to be $Var[X] = 4$.

So, $$E[(X-t)^2] = Var[X] = 4$$

Answer 2

Since $Y = \frac{X - t}{2},$ therefore $X - t = 2Y,$ therefore $(X - t)^2 = 4Y^2.$

And you already know $E[Y^2].$