Homework 8 Answers

BSTA 550

Chapter	Turn In	Extra Problems
25	TB # 18, NTB # 1	# 1, 4, 8, 17, 23, 24 Slide examples: 2, 3.3, 4
26*	TB # 12**, NTB # 2, 3	# 7, 9, 19, 20 Slide examples: 3
27	TB # 12***	# 6, 8, 13, 17 Slide examples: 1.2
28	TB # 18	TB # 1, 10
29	NTB #4
30		TB # 4, 7-12

* Although within Chapter 26, these exercises are primarily practicing the material from Chapter 25.

** For this problem, you only need to set up the integrals!!

*** For Ch 27 # 12, in order to find the conditional densities in parts (a) and (b), you will need to calculate \(f_Y(y)\) for the specific regions of \(y\) specified. After finding the conditional densities in parts (a) and (b), also calculate the conditional probabilities below. Please submit these together with your other work in parts (a) and (b): Find \(\mathbb{P}[0.5 < X < 3 | Y = 4]\). Find \(\mathbb{P}[0.5 < X < 3 | Y = 7]\).

Non-textbook problems

• #1: \(f_Z(z) = nF_X(z)^{n-1}f_X(z)\)

• #2: (b) \(P(X<Y) = 0.099995\) (d) \(f_Z(z) = \dfrac{11-2z}{5}e^{-2z}\) for \(0 \leq z \leq 5\)

• #3: 2 cases: when \(0< z < 1\) and \(1 \leq z \leq 2\)

• #4: (a) 63 minutes, (b) \(Var(T) = \dfrac{287}{3}\), (c) \(Var(D) = \dfrac{41}{3}\), (d) expected diff is 7 minutes, variance is 283/3

Textbook problems

There are answers at the back of the book!! Selected answers (or hints) not provided at the end the book:

• Chapter 25

  • # 4:   7/16

  • # 8:  (a) \(\frac{25}{228}\)     (b) \(f_X(x)=\frac{1}{12}(x+1)\), for \(0\leq x\leq 4\)     (c) \(f_Y(y)=\frac{3}{76}(y^2+1)\), for \(0\leq y\leq 4\)

  • # 18:  5/6

  • # 24:  (a) \(f_X(x)=-2e^{-2x}+2e^{-x}\), for \(x\geq 0\)     (b) \(f_Y(y)=2e^{-2y}\), for \(y\geq 0\)

• Chapter 26

  • # 12:  (b) \(\frac{233}{256}\)     (c) \(\frac{65}{256}\)     (d) \(\frac{1}{512}\)

  • # 20:  (a) Yes.     (b) \(\frac{15}{16}\)

  • NTB # 3: (b) 0.09999546   (d) \(f_Z(z) =\Big(\frac{11}{5} - \frac{2z}{5}\Big)e^{-2z}\), for what values of \(z\)?

• Chapter 27

  • # 6: \(f_{X|Y}(x|y)=\frac{e^{-x/4-y/5}}{4(e^{-y/5}-e^{-9y/20})}\), for \(0< x< y\)

  • # 8: \(f_{X|Y}(x|y)=\frac{1-x^2}{1-y-\frac{(1-y)^3}{3}}\), for \(0\leq x, 0\leq y, x+y\leq 1\)

  • # 12: (a) \(f_{X|Y}(x|y)=\frac{1}{2}\)    (c) \(\frac{4}{7}\)

• Chapter 28

  • # 10: (a) 8/9     (b) 14/3     

  • # 18: 4/5

• Chapter 29

  • # 10: (a) 26/81     (b) 74/9

  • # 14: (a) 67/3     (b) 1/14     (c) 25/12     (d) \(\sqrt{25/12}\)

  • # 26: 250

  • # 32: See notes (or book) for the proof from the discrete random variables case. The proof doesn’t depend on what type of random variable (discrete vs. continuous) is being used.

  • NTB # 3: (a) 63     (b) 287/3     (c) -1, 41/3     (d) -7, 287/3

• Chapter 30

  • # 4: \(f_x(x)=1/2\) for \(2\leq x\leq 4\)

  • # 8: (a) T     (b) T     (c) F

  • # 10: (a) F     (b) T

  • # 12: (a) T     (b) T     (c) F     (d) T