M241 Probability Semester 011

Reading Assignments

Chapter 1
Chapter 2 and Appendix 1
Chapter 3
Chapter 4
Chapter 5
Chapter 6

Homework Assignments

Chapter 1
Chapter 2 and Appendix 1
Chapter 3
Chapter 4
Chapter 5
Chapter 6

Reading Assignments

• Section 1.1, 1.2 Wed. Aug. 22nd
• Section 1.3 Friday Aug. 24th
• Section 1.4 Monday Aug. 27th
• Section 1.5 Wednesday Aug 29th
• Section 1.6 Friday Aug 31st

• Section 2.1  and Appendix -- Friday September 7th
• Section 2.2  Wednesday September 12th
• Section 2.4  Friday September 14th
• Section 2.5  Monday September 24th

• Section 3.1  Wednesday September 26th
• Section 3.2  Monday October 1st
• Section 3.3  Wednesday October 3rd
• Section 3.4 Friday October 5th
• Section 3.5 Monday October 7th

• Section 4.1   Wednesday October 31
• Section 4.2   Friday November 2nd
• Section 4.5   Monday November 5th

• Section 5.1 through page 343  Monday November 12th
• Section 5.2 Wed. November 14th

• Section 6.1, 6.2
• Section  6.4

Homework Assignments

Chapter 1

For Wednesday August 22nd

• Toss a coin 200 times (honest)
• Record in order H or T that comes up.
•  If you are supposed to be a cheater,  don't really toss the coin -- just write down H or T randomly 200 times.
• Bring results to class.

 

• Exercises 1.1: 1, 2, 4, 5, 6, 7a, b, c, 8 a, b, c
• Exercises 1.2: 2

• Exercises 1.3: 2, 3, 6, 8, 9, 10
• Exercises 1.4: 2, 3, 4, 5, 6, 7, 8
A few hints on these exercises if needed:
1.3 For 8, 9, and 10 you must use various combinations of the rules of probability  in this section -- Note there is a nice summary of these on page 72  --the  last three are most important in these problems.  Draw Venn Diagrams and label known probabilities. Then plug in known information into the formulas.
1.4:  For number 8  -- use the rule for conditional probabilties  --  but remember that the while 50% of the cards are black on one side and white on the other side, only  half of these (25% total) after random mixing will have the white side up and black side down.

• Exercises 1.5:  2, 3, 4, 5

• Exercises 1.6:  1, 2, 3, 4, 7

Chapter 2 and Appendix

Appendix 1 -- page 514:  x, xi, xii, xiii
Section 2.1 1, 2, 3, 4, 5, 6, 7, 9, 12

Exercises 2.2  2a,b, 4, 9

Exercises 2.4:  2, 3, 5

Exercises 2.5: 1, 2, 3, 4

Chapter 3

Exercises 3.1:  1, 2, 3, 4, 5, 6
Exercises 3.2:  1, 3

• Exercises 3.2:   4, 5, 6, 8, 9, 11, 13

Hints for Section 3.2:
Hint on 4:  Show that if 26 or more numbers are 8 or more, the average would have to be greater than 2.
Hint on 6:  Use the Indicator Method (like example 8)
Hint on 8 and 9:  First expand (X+Y)^2  or (X-Y)^2
Hint on 11: Use Markov's Inequality
Hint on 13:
Review example 9, problem 2 for part b)  -- Use E(sume of largest two) = E(Sum of all 3 - Minimum).
For part c) P(Max = m) = P(Max <= m) - P(Max <= m-1), then definition of Expectation  to calculate E(Max).
For part d)  Note that the n umber of multiples of 3 in the first 10 rolls will have binomial distribution with n = ____ and p = ____.
For part e)  Use the method of indicators.   Let I_i = indicator of the event that face i fails to appear in the first 10 rolls.
For part f)  Note that the number of DIFFERENT faces in the first ten rolls = 6 - the number of different faces that fail to appear, and your result from part e).

• Exercises 3.3:  1, 2, 3, 5
 
Hint for Section 3.3 Number 5:  Use computational formula for variance to compute Var(X-a) and also that Var (X-a) = Var(X).

• Exercises 3.3:  12, 14, 16, 18

Hints:
For 12 --  X is not assumed to be Normal -- So you will have to use Chebyshev's inequality.
For 14 use Markov's and Chebychev's inequalities.
For both 16 and 18 you will be approximating the sum of independent and identically distributed random variables with the Normal distribution. You will have to standardize the sum to use the Normal (0,1) table -- as in example 9.

• Exercises 3.4:  1, 2, 3a -- as indicated  Also do b)  Same as stated except waiting time until 4th success.
• Exercises 3.5:  2, 4, 8, 11 (for 11 note that the sum of Poisson r.v's is also Poisson with parameter = ____ )

Chapter 4

• Exercises 4.1   1, 2, 3, 6, 7, 8

Hints
1.  For this problem approximate with f(x)dx since dx is small.
6. Hint: Want P[X<=0] = P[X* <= (0-m)/s] = 1/3
and P[X <=1) = P[X* <= (1-m)/s] = 1. Use normal table to set up equations that you can solve for m and s.
7. Hint: Mean is 5'10". Use the fact that 10 percent of pop is over 6' tall and the Normal Table to calculate what the standard deviation must be. Next calculate the probability that an individual picked at random is over 6'2" using the Normal table. Finally the chance of 2 least successes in 100 trials of picking people at random is Binomial (n,p) where n = ____ and p = _____ -- This  is well-approximated by the Poisson distribution (because p is small  -- with mean = np).
8.  Hint a) Just use Normal distribution table
b) You are finding P[ 11.8 < S₁₀₀/100 <12.2] which is Normal ( , ) (review pages 194-196)

Exercise Set 12  Due Friday November 9th

Exercises 4.2 1,3,4,5

• Hint on 1 c)  Think Binomial.  Each atom can either survive longer than time t (success) or not survive  longer than time t   (failure).  N_t =  number of atoms out of 1024  surviving for at least t years is a binomial distribution with n = 1024, and p = P(T>t) = .......   Want to find t such that E( N_t) = 1.
• Hint on 1d) Approximate the Binomial distribution using the Poisson(np) distribution.
• Hint on 4b)  Median m says want P(W>m) = 1/2;  solve for m.
• Hint on 4d)  Let X = average of 100 independent and identically distributed random variables so E(X) = _____ and Var(X) = ________ -- review page 194. X is well approximated by the Normal distribution by the Central Limit Theorem so now just plug and chug.

Hint on 4e) This is the Gamma Distribution with r = 2  --  P(W1 + W2 > 11)  = P( Number of failures is less than 2), where number of failures is Poisson with parameter _____.


Exercise Set 11:  Due Wed November 14th
 Exercises  4.5  1, 2, 3, 6, 8

Hints:
1.  We will do part a in class  -- just simple integration
2.  These are step function since discrete -- see examples in this section.
          Sketch and also write down what F(x) is at each point that it jumps
          For part b) note that the sum simplifies (see appendix page 516 bottom).
          3 follows from example 2 in text a) is just a simple observation of the symmetry.
          b)  Draw the sketch, then calculate the probability as a proportion of the unit
circle  area.
          6a), b), and c)direct calculations  from definitions.d)  See page 316 -- X is the
          maximum {Y1, Y2, Y3}

Chapter 5

• Section 5.1 Exercises 1, 3, 4, 6
Hints for 5.1:   Draw sketches for each problem.
1. Draw sketch of the range of (X,Y) = R = {(x,y): ) < x < 2 and 0 < y < 4 and x < y}
For a) P(X < 1) = Area of region in R where (X < 1) / Area of R
For b) P(Y < X²) = Area of region in R where (Y < X²) / Area of R
3. Do this like Example 1, problem 1 of this section.
4. Do this like Example 2, a,b,c of this section
6.
a) Set up similarly to Example 3 of this section. Calculate P(X < Y - 2)
b) Let F = event that first arrives before 12:05.
Let L = event that last person arrives after 12:10.
Review some of the set identities from Chapter 1 first week of class!!!
We want to calculate P(F intersect L). Use the following 3 facts:
• P(F union L) = P(F) + P(L) - P(F intersect L)
• P(F union L ) = 1 - P((F union L)^c)
• (F union L)^c = F^c intersect L^c
to get a formula for P(F intersect L) in terms of P(F), P(L) and P(F^c intersect L^c)
Then calculate P(F), P(L), and P(F^c intersect L^c). Do this by first writing out what each of these sets means.
For example P(F) = P(first arrives before 12:05) = 1 - P(F^c), where F^c is the events all 10 arrive at time >= 12:05. But by  independence P(F^c) = P(X₁>12:05)…P(X₁₀> 12:05) = (10/15)¹⁰
 
Exercise Set 13  Section 5.2  Exercises 3, 4, 5  Due Wed November 28th
Hints for Exercises Section 5.2
3.  a)  Integrate the joint density function over the region on which it is defined.  What value of c gives a value of 1 for this integral?  b)  Integrate the joint density function over the regtion 0 < x < a and 0 < y < 1. c) Similar to b)
4.  a)  Just integrate the joint density function first with y going from 0 to y  (you might not like this notation) and then for x going from 0 to x. b)  Integrate the joint density funciton with respect to y.
5.  Almost identical to example 2 in this section.

Chapter 6

Exercise Set 13  Due Fri. December 7th
Section 6.1:  1
Section 6.2:  1
Section 6.4:  5, 6