How to evaluate Probability of Y? 
Hi all,
It's my first undergraduate statistics module as a business major and I've encountered some difficulties in computing the response to the question.
I have several queries below:

*

*Would Y have to represented in terms of Z?

*Would it be better to perform the summation or expansion of the Xj term first?

*Would the Expectation of Y and Variance of Y be required to compute P(Y > 1.61)?

*Wouldn't the Expectation of Y = 0 and Variance of Y = 1 since its variables are Z, which is a standard normal variable?

*Are there 2 different answers for P(Y > 1.61)?

Many thanks in advance and cheers!
 A: Thanks for reply with your thoughts. You're getting closer.
After a bit of simplification, I got $Y = \sum_{j=1}^5 Z_i^2 \sim \mathsf{Chisq}(\nu = 5).$
Then using R instead of a printed table of chi-squared distributions, I got
$P(Y > 1.61) \approx 0.90.$
1 - pchisq(1.61, 5)
[1] 0.9000374
qchisq(.1, 5)
[1] 1.610308

Notes: (1) In R, pchisq is a chi-squared CDF, and qchisq is an inverse CDF (quantile) function.
(2) The chi-squared distribution with $\nu$ 'degrees of freedom' is defined as the distribution of the
sum of squares of $\nu$ independent standard normal random variables.
(3) If $Q \sim \mathsf{Chisq}(\nu),$ then $E(Q) = \nu,$ $Var(Q) = 2\nu.$
(4) Density function of $\mathsf{Chisq}(5).$

curve(dchisq(x,5), 0, 20, lwd=2, ylab="PDF", xlab="q", 
      main="Density of CHISQ(5)")
 abline(v=0, col="green2"); abline(h=0, col="green2")
 abline(v = 1.61, col="red", lty="dotted")

(5) Simulation of $\mathsf{Chisq}(5)$ as the sum of squares of five
standard normals.
set.seed(2020)
q = replicate(10^5, sum(rnorm(5)^2))
mean(q)
[1] 4.984788  # aprx E(Q) = 5
var(q)
[1] 9.977126  # aprx Var(Q) = 10

hist(q, prob=T, br=30, col="skyblue2", 
     main="Simulated Dist'n of CHISq(5)")
     curve(dchisq(x,5),add=T,lwd=2,col="brown")


