Homework Assignments


# Problems Due Notes
1
  • Section 12.1: 1, 5, 7, 11, 13, 15, 23, 30 (Ans: 2), 39.
    (Answers to odd numbered problems can be found in the last pages of the book).
  • How many ternary vectors of length n are there? (a ternary vector of length n is a sequence of n digits, each being 0,1 or 2. For example, 012120 is a ternary vector of length 6).
    (Ans: If n=5 the answer is 243).
  • How many words are there with 3 'A's, 5 'B's, 2 'C's and 1 'D'. (Ans: 27720).
1/10/2011
  • Please attach the cover page and staple all your pages.
  • Justify your final answer. Wrong or not enough justification will be scored poorly.
2
  • Section 12.2: 17.
  • Section 12.2: 19, 21.
    For these two problems also:
    • Describe a suitable equi-probability sample space.
    • Describe the events explicitly as subsets of these space.
    • Describe the following events explicitly and find their probability:
      • "At least one 4 *and* the first number is greater than the second number" (Ans: 5/36).
      • "At least one 4 *or* the first number is greater than the second number" (Ans: 7/12).
      • "None of the dice show 4" (Ans: 25/36).
  • The numbers 1,2, ..., 10 are randomly ordered in a circle. What is the probability that when going clock-wise, you can find the sequence 1,2,3,4,5 in the circle. Find a sample space explicitly and describe the event as a subset of it (Ans: 5!/9!).
  • Section 12.2: 31, 33, 35.
  • Section 12.2: 9,10 (Ans: 0.75, 0.3), 11, 12 (Ans: 1), 13, 15.
1/19/2011
  • Due to holiday on 1/17, homework is due on 1/19.
3
  • Section 12.3: 1,3,5,7,9,11,13,15,19.
1/26/2011
  • Due to late posting of HW3 and the midterm on Monday, submission is postponed to this date.
4
  • Section 12.3: 31,33.
  • In a given population, there is an equal number of women and men, but the number of color blind men is 4 times higher than the number of color blind women. A person is chosen randomly and it turns out that he/she is color blind. What is the probability that it is a man? (Ans: 4/5) .
  • Section 12.3 (yes 12.3 again): 21, 23. Repeat problem 23, but this time assume that the balls are replaced (Ans: Independent) .
  • Assume that the chances of giving birth to a boy are the same as those of giving birth to a girl and that succeeding births are independent. A family is randomly selected. Let A be the event that it has at least one boy and at least one girl. Let B be the event that it has at most one girl. Show:
    • A and B are dependent if the family has 2 children.
    • A and B are independent if the family has 3 children.
  • Section 12.4: 1, 3, 9. For problem 9, also find P(X greater or equal than 8) (Ans: 27/25) , P(X is even) (Ans: 6/11) .
1/31/2011
  • Note that this HW is due on a Monday (as usual)
5
  • 12.4.29. Also find P(X = 5 | X < = 9) (Ans: 0.246) .
  • 12.4.31, 12.4.33. For 12.4.33, also find the probability that exactly 3 carried aphids (Ans: 0.205) and the probability that exactly 3 carried aphids given that at least one carried aphids (Ans: 0.207) .
  • 12.4.39, 49, 53, 55, 59. For 12.4.59, assume now that the urn contains 3 blacks and 9 whites. Repeat (a) and (b) (Ans: (a) 0.059, (b) 0.068) .
  • There are 1000 fish in lake, among which 2 are gold fish. Dan took out 1000 fish with replacement. Which event is more likely: A = "Less than 2 gold fish were found". B = "More than 2 regular fish were found". (Ans: B)
  • You toss a coin until the first time you get the same side as the one in the first toss. Let X be the number of tosses. Find the p.m.f of X. Is X a Geometric R.V.? (Ans: Not geometric, for example P(X=5)=1/16) Now assume that the coin is biased and that the probability of getting heads is 2/3. Find the p.m.f of X on this case (Ans: for example, P(X=5) = 12/243) .
2/7/2011
6
  • 12.4.63, 12.4.65, 12.4.73, 12.4.77, 12.4.79, 12.4.81, 12.4.83. For 12.4.81, also find the probability that exactly 4 people experience side effects. (Ans: 0.0016) For 12.4.83, also find the probability (with and without the approximation) that there are at most five cases. (Ans: 0.994(with))
  • The percentage of left-handed students is 1.5%. In a lecture hall there are 100 seats, among which 3 are designated for left-handed.
    • Approximate the probability that in a full lecture hall, there will not be enough seats to accommodate all left-handed students. (Ans: 0.066) .
    • Every week 50 lectures are given in that hall. What is the probability (precisely) that in at least one of these lectures there were not enough seats for the left-handed? (Ans: 0.967) .
    • There are currently 100 students in the hall, 3 of which are left-handed. 5 students are chosen randomly in the room. Find the probability that exactly one of them is left-handed, both precisely (Ans: 0.138) and using the Binomial approximation (Ans: 0.133) .
  • Find the expected value of the random variables whose p.m.f. appear in problem 12.4.5 (Ans: -0.1) and 12.4.15.
  • Find the expected number of Hearts in a random poker hand. (Ans: 1.25)
  • Find the expected sum of the values shown by two dice, rolled together. (Ans: 7)
  • Find the expected number of heads in a sequence of 4 tosses of a fair coin. (Ans: 2) Repeat if the coin gives heads with probability 1/3. (Ans: 1.33)
2/14/2011
7
  • 12.4.15. Also let Y=X^4 (X to the forth power) and Z=3X-1. Find:
    • The p.m.f. of Y and Z. (Ans (partial): P(Y=1)=0.6, P(Z=-1)=0.3)
    • E(Y) and E(Z), once using the p.m.f's above and once using the formula for the expectation of a function of a r.v. (summing f(x)P(X=x) for all x). Make sure to get the same answer. (Ans: EY=1.2, EZ=-2.2)
    • Is E(X^4) = (E(X))^4? (Ans: No) Is E(3X-1) = 3E(X)-1? (Ans: Yes.) Explain the answer you got. (Ans: E(aX+b) = aE(X)+b))
  • 12.4.17.
  • 12.4.19. Repeat with X being uniformly distributed on the set {1, 2, ... 20} . (Ans: E(X) = 10.5, Var(X)=33.25) Do you get a smaller or larger variance? (Ans: Larger) Explain why you get this answer.
  • 12.4.19, 12.4.21, 12.4.25.
  • Find E(X), Var(X), SD(X) and E(Y), Var(Y), SD(Y) of the following:
    • X ~ Bin(n=10, p=1/3). Y=2X+3. (Ans: E(X)=3.33, V(X)=2.22, SD(X)=1.49, E(Y)=9.67, V(Y)=8.88, SD(Y)=2.98)
    • X ~ Geo(p=1/4). Y=10X.
      • (Ans: E(X)=4, V(X)=12, SD(X)=3.46, E(Y)=40, V(Y)=1200, SD(Y)=34.6)
    • X ~ Ber(p=1/10). Y=-X+2.
      • (Ans: E(X)=0.1, V(X)=0.09, SD(X)=0.3, E(Y)=1.9, V(Y)=0.09, SD(Y)=0.3)
    • X ~ Pois(10). Y=-3X-1.
      • (Ans: E(X)=10, V(X)=10, SD(X)=3.16, E(Y)=-31, V(Y)=90, SD(Y)=9.48)
    • X ~ HG(N=100, G=20, B=80, n=10). Y=2X+1.
      • (Ans: E(X)=2, V(X)=1.29, SD(X)=1.14, E(Y)=5, V(Y)=5.16, SD(Y)=2.28)
  • 12.4.5, 12.4.7.
  • You choose a point "uniformly at random" in the square whose vertices are (1,1), (0,2), (-1, 1), (0,0). Let (X,Y) be the x and y coordinate of the chosen point.
    • Find the c.d.f. of X and Y. (Ans: F(x) is 0 if x < -1, ((1+x)^2)/2 if -1 < x < 0, 1-((1-x)^2)/2 if 0 < x < 1 and 1 if x > 1. For F(y), replace x with y-1 in the above)
    • Graph them.
    • Show that these c.d.f's have the general properties of c.d.f.'s discussed in class (e.g. non-decreasing).
2/23/2011 Submission postponed by two days, due to academic holiday on 2/21.
8
  • 12.5.1. Also find: P(X<2), P(X>=2) , P(X>2), P(X=2), P(1 < X < 2). (Ans: 0.9975, 0.0025, 0.0025, 0, 0.047)
  • 12.5.3. Also find: P(X<-3), P(X>=-3), P(X>-3), P(X=-3), P(-3 < X < -1). (Ans: 0.102, 0.898, 0.898, 0, 0.148)
  • For the functions below, check whether they could be a c.d.f. for a continuous random variable X and if so, find the density of X (as usual a^b means a to the power of b):
    • F(x) = exp(-e^(-x)). (Ans: Yes, f(0) = 1/e, f(1) = 0.255)
    • F(x) = x/(x+1) if x >= 0 and 0 if x < 0. (Ans: Yes, f(-1)=0, f(2)=1/9)
    • F(x) = 1-(1-e^(-x))/x if x > 0 and 0 if x <= 0. (Ans: Yes, f(-1) = 0, f(1) = 0.264)
  • Find the density, expectation and variance of X and Y in the last problem in HW 7. (Ans: X: f(-0.5) = f(0.5) = 0.5, E(X)=0, Var(X)=1/6. Y: f(0.5) = f(1.5) = 0.5, E(Y)=1, Var(Y) = 1/6)
  • 12.5.5, 12.5.7, 12.5.9.
  • Let Z be the distance to the nearest edge of a point uniformly chosen in the interval [0,1].
    • Find the c.d.f, density and expectation of Z. (Ans: F(0.1)=0.2, F(0.4)=0.8, f(0.1)=2, f(0.4)=2, E(Z)=0.25.)
    • Find P(Z < 0.2). (Ans: 0.4)
    • Find E(Z^3). (Ans: 1/32)
    • Let Y be the number of points whose distance to the nearest edge is less than 0.2, when 25 points are independently randomly selected in the interval [0,1]. Find P(Y = 10). (Ans: 0.161)
  • The precentage X of correct answers in a midterm has the following density:
    f(x) = cx(100-x) if 0 <= x <= 100 and 0 otherwise.
    • Find c and the probability of failing (getting less than 55). (Ans: 6*10^(-6), 0.575)
    • 5 students took the midterm. Find the probability that the majority will pass. (Ans: 0.361)
    • Calculate E(X), E(X^4). (Ans: 50, (10^8)/7)
2/28/2011
9
  • 12.5.37, 12.5.39, 12.5.41, 12.5.45.
  • 12.5.49.
  • 12.5.51. Assume now that the lifetime T of the atom is Uniformly distributed in the interval [0,54] (so that its mean is still 27). Repeast (a) and (b). Does the lack of memory property hold for the Uniform distribution? (Ans: 0.63, 0.412, no)
  • 12.5.53.
  • 12.5.55. Assume now that you have bought the device from someone and you do not know how long he has used it before. What is the probability that the device will have failed after three years from your purchase? (Ans: 0.451)
  • Your lamp uses 5 light bulbs to operate. Suppose that each bulb has a lifetime which is exponential with mean 1 year, that the lifetimes are independent random variables and that the lamp does not light if at least one of the bulbs is burnt.
    • What is the distribution of the time until the lamp stops working? (Ans: exp(5))
    • Find the probability that the lifetime of the entire lamp is more than a year. (Ans: 0.007)
  • The half-life of a radioactive atom is a time t such that the probability that the atom did not decay by time t is 0.5. It is known that the time until an atom decays is Exponentially distributed. Strontium-90 is a radioactive atom which has a half-life of 28 years.
    • Find the rate and expected value of the time until Strontium-90 decays. (Ans: 0.025, 40.395)
    • Assume that you have 2gr of S-90 (each 1gr has 10^22 atoms). What is the expected mass which will remain after 100 years? (Ans: 0.168)
    • How long do you have to wait, until the expected mass is less than 1% of what it was? (Ans: 186.03)
3/7/2011
10
  • 12.6.15 - 12.6.37
N/A