STA 257 (Fall 2012): Probability and Statistics I

Below is a list of good practice exercises from the text. Do as many as you reasonably can. Bring questions about any of them to the stat aid center, or for tutorial discussion.

Assignments

• Problem Set 1 (for discussion and quizzing at tutorial Sept 19, but not for hand-in)
  Read Lecture 1 and sections 1.1-1.4 from the textbook.
  Practice problems: p. 26-30, # 1, 2, 4, 5, 7, 8, 11, 12, 15, 20, 21, 26, 27, 30, 36, 38, 41, 42.

• Problem Set 2 (for discussion and quizzing at tutorial Sept 26)
  Read Lecture 2 and sections 1.5-1.7 from the textbook.
  Practice problems: p. 30-34, # 45, 47, 48, 49, 51, 56, 57, 62, 68, 71, 73, 77, 78.
  
  Extra problem: If A is an event then the indicator function (rv) of A is defined as I(A)(x)=1 if x is in A and I(A)(x)=0 if x is not in A. I(A) is an example of a discrete rv. It is also called a Bernoulli random variable.
  Show that I(AUB)=I(A)+I(B)-I(AB) . (AB = intersection of A and B)
  
  
• Problem Set 3 (for discussion and quizzing at tutorial Oct 3)
  Read Lectures 3 and 4 and sections 2.1 and 2.2 from the textbook.
  Practice problems: p. 64-69, # 1, 2(b), 3, 4, 6, 7, 11, 12, 19, 25, 29, 30, 31, 33, 35, 39, 40, 45, 46, 48, 49, 52, 53.
  
  Problem Set 3
  Answers to some even numbers:
  • #2 (b): p(0)=5/16, p(1)=6/16, p(2)=4/16. p(3)=1/16; F(0)=5/16, F(1)=11/16, F(2)=15/16, F(3)=1
  • #12: 9 heads in 10 tosses
  • #30: (a) p(k)=mu^k exp(-mu)/k!, mu=3.96, 3 is most probable; (b) 0.003
  • #40: (a) c=3; (b) F(x)=x^3; (c) 0.124
  • #46: 0.0513
  • #52: (a) 25%; (b) N(177.8, 58.1), N(1.78, 0.0058)
  
  
• Problem Set 4 (for discussion and quizzing at tutorial Oct 10)
  Read Lecture 5 and sections 2.3 and 4.1, 4.2 (for the univariate case only) from the textbook.
  Practice problems: p. 69 # 57, 58, 62, 63, 67, 68 p. 166 - 169 # 2, 3, 5, 7, 14, 16, 30, 31, 34

  Problem Set 4
  Answers to some even numbers:

  • p. 69 # 58: f(v)=2v, 0<=v<=1
  • #68: f(x)=1/(2sqrt(pi x)) exp (-sqrt(x/pi))
  • p. 166 - 169 #2: E(X)=(n+1)/2, Var(X)=(n^2-1)/12
  • #14: (a) 2/3; (b) 1/2, 1/18
  • #30: 1/lambda(1-exp(-lambda))
  • #34: 2/3
  
  
• Problem Set 5 pdf (for discussion and quizzing at tutorial Oct 17).
  
  
• Problem Set 6 (for discussion (Oct 24/31) and quizzing at tutorial Oct 31)
  
  • Read Lecture 7 and sections 3.1 - 3.5 from the textbook.
  • Practice problems: p. 107-110 # 1, 2, 3, 7, 8, 9, 12, 14, 20, 27.
  
  
• Problem Set 7 (for discussion and quizzing at tutorial Nov 7)
  • Read Lecture 8 and sections 4.3 - 4.4, 5.1 - 5.2 from the textbook.
  • Practice problems: p. 169-173 # 42, 43, 44, 47, 50, 55, 56, 57, 59, 63, 70, 75, 77 (a,b);
  • p. 188 # 1, 2.
  
  
• Problem Set 8 (for discussion and quizzing at tutorial Nov 14 and Nov 21)
  • Read Lecture 9 and sections 3.6, 6.1 - 6.2 from the textbook.
  • Practice problems: p. 112-113 #43, 46, 48, 50, 51, 52 ( assume both r.v.'s are uniform (0,1)), 58 (optional), 61, 63, 64;
  • p. 198 # 5, 6, 7, 8.
  
  
• Problem Set 9 (for discussion and quizzing at tutorial Nov 28)
  • Read Lecture 10 and sections 3.7, 4.5 from the textbook.
  • Practice problems: p. 114 # 69, 73, 79; p. 173-174 # 79, 80, 83, 85, 86, 88, 89, 91.
    
    
  • Please use the following formula where needed (you don't need to derive it)
    
  
  
• Problem Set 10 (there is no tutorial for that week, so please use TAs' office hours to discuss the problems)
  • Read Lecture 11 and section 5.3 from the textbook.
  • Practice problems: p. 188-190 # 3, 4, 5, 6, 10, 16, 17, 18.
  
  
• Problem set 10:
  Answers to even numbers:
  • #4: delta=16.45
  • #6: standardize gamma variable by subtracting its mean and dividing by st. deviation, then apply Taylor's expansion to the mgf of the standardized variable.
  • #10: 0.48, 0.002
  • #16: P(S<=10)= appr 0
  • #18: 0.02

sta257 (Fall 2012): Probability and Statistics I || http://www.utstat.toronto.edu/~olgac/sta257_2012/