Probability Distribution 22

Question

X1 ∼ N(µ = 2, σ = 2 X2 ∼ N(µ = 1, σ = 4 X3 ∼ N(µ = −4, σ= 3 X1, X2, and X3 be independent and Y = (X1 + 2X2 + X3)^2

Determine P(Y > E(Y ))

My solution:

I got the value of E(Y) as 53

So, P((X1+2x2+x3)^2 >53 )

Can I get some hints on how to proceed further?

Answer 1

You should be able to find the distribution of $W:=X_1+2X_2+X_3$ (they are a sum of independent normals).

Now use the fact that for $a\geq 0$, $\{W:W^2>a\}= \{W:W<-\sqrt a\}\cup \{W:W>\sqrt a\}.$