Math 5080, Fall 2020
Homework 5
Version: September 28, 2020 7:02pm Z
Due Monday, October 5th at 11:59 PM
Instructions (Read before you begin)
• Turn in solutions to the questions marked in bold red
• Only one of these questions will be graded and is out of 10 points
• Upload solutions directly to Canvas
• All uploads must be in PDF format so that I can mark them directly
• I have also included the TeX file if you want to type up your solutions, but this is not required
• Question numbers are from the Bain and Englehardt book, unless otherwise indicated
8.8 Suppose that X ∼ χ2(m), Y ∼ χ2(n) and X and Y are independent. Is Y −X ∼ χ2 if n > m?
If so what are the degrees of freedom?
8.9 Supposes that X ∼ χ2(m), S = X + Y ∼ χ2(m + n), and X and Y are independent. Is
S −X ∼ χ2(n)?
8.14 If T ∼ t(ν), give the distribution of T 2.
8.15 Suppose that Xi ∼ N(µ, σ2) for i = 1, 2, . . . , n, and Zi ∼ N(0, 1) for i = 1, 2, . . . , k, and that
all variables are independent. State the distribution of each of the following variables if it is
a “named” distribution, or otherwise state “unknown”.
(a) X1 −X2
(b) X2 + 2X3
(c) X1−X2
σSZ
√
2
, where SZ indicates the sample standard deviation of the Z terms
(d) Z21
(e)
√
n(X̄ − µ)/(σSZ)
(f) Z21 + Z
2
2
(g) Z21 − Z22
(h) Z1
√
Z22
(i) Z21/Z2
1
(j) Z1/Z2
(k) X̄/Z̄
(l) √
nk(X̄ − µ)
σ
√∑k
i=1 Z
2
i
(m)
∑n
i=1
(Xi−µ)2
σ2
+
∑k
i=1(Zi − Z̄)2
(n) X̄/σ2 + 1k
∑k
i=1 Zi
(o) kZ̄2
(p)
(k − 1)
∑n
i=1(Xi − X̄)2
(n− 1)σ2
∑k
i=1(Zi − Z̄)2
8.18 Assume that Z, V1, and V2 are independent random variables with Z ∼ N(0, 1), V1 ∼ χ2(5),
and V2 ∼ χ2(9). Find the following
(a) P(V1 + V2 < 8.6)
(b) P(Z
√
V1/5 < 2.015)
(c) P(Z > 0.611
√
V2)
(d) P(V1/V2 < 1.450)
(e) The value b such that P
(
V1
V1+V2
)
= 0.90
8.19 If T ∼ t(1) then show the following:
(a) The CDF of T is F (t) = 1/2 + arctan(t)/π
(b) The 100× γth percentile is tγ(1) = tan[π(γ − 1/2)]
2