I cannot read that scan.
I bet I can answer your questions if you would retype it for me.
It looked like a linear combination of normals and also squares of normals, creating Chi-Square rvs.
If so, that should be rather easy.
There might be some t and F's in there too.
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I blew it up and I had to laugh.
I'm teaching out of Wackerly (I also knew Dennis when I was at Florida)
this semester and I just assigned those two problems this year.
I even have the sol manual, but thats not necessary
37a each Y_i is a N(0,1)
so the square is a Chi-square with one df
the sum of indep chi-squares is a chi-square with the df each to the sum
of those df so the answer is chi-square with 5 dfs
37b U divided by sigma squared is a chi-square w n-1=4 df
but since sigma=1 here we don't need to see sigma.
Hence we have a chi-square w 4 dfs.
37b Thats U (from part b) plus an indep chi-square w 1 df
hence it's a chi-square w 5 df (add the dfs)
38a that's a t with 5 dfs, because its a standard normal divided
by a chi-square divided by its dfs and square rooted
38b that's a t with 4 dfs, for the same reason as in (a), just take
the 2 and rewrite him as square root of 4 and place him under the denominator inside a square root.
38c thats an F with 2 and 4 dfs, because you can write him as a chi-square (with 2 df)divided by two OVER a chi-square (with 4 dfs) divided by 4.