I just began to take this stats course in HS and I'm a little stuck on these 2 problems below. Can anybody please help me out with the solutions? Thank you. Anything is appreciated.
Let $Y$ be a random variable. Define a function $Q$ by $Q(m) = E[(Y − m)^{2}]$
(a) Write $\mu E[Y]$. How do you show that $Q(m) = \text{var}[Y] + (m − \mu)^{2}$. I was thinking it had something to do with $Y −m=(Y −\mu)−(m−\mu)$.
(b) Also, how do you show that $Q(m)$ is minimized at $m = E[Y]$. I was thinking that it had something to do with checking the sign of $Q(m) − Q(\mu)$.
Let $Y$ be a random variable distributed with $N(3, 16)$. Find $Pr[Y \gt 9.2]$.
You are right about the approach. We have $Y-m=(Y-\mu)+(\mu-m)$ and therefore $$(Y-m)^2=(Y-\mu)^2+2(\mu-m)(Y-\mu)+(\mu-m)^2.$$ Now take the expectation, noting that $E(Y-\mu)^2=\text{Var}(Y)$. By linearity we get $$E(Y-m)^2=\text{Var}(Y)+2(\mu-m))E(Y-\mu)+E((\mu-m)^2).$$ Now we are finished, because $E(Y)=\mu$, so $E(Y-\mu)=0$, and $\mu -m$ is constant, so $E((\mu-m)^2)=(\mu-m)^2$.