Homework Assignments


# Problems Due Notes
1
  • Section 12.1: 1, 9, 17, 25, 33, 41, 49.
    (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).
  • Section 12.2: 1, 5, 9, 13, 17, 21.
  • Section 12.2: 25, 27.
    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, 35, 39, 43, 47, 51.
6/28 Please don't forget to attach the cover page and staple!
2
  • Section 12.3: 1, 5, 9, 13, 15, 19, 21, 25, 27, 29. Repeat problem 29, but this time assume that the balls are replaced (Ans: Independent) .
  • Section 12.3: 31, 33, 35, 39 .
  • 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) .
  • 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, 5, 7, 13. For problem 13, also find P(S greater or equal than 8) (Ans: 27/25) , P(S is even) (Ans: 6/11) .
7/12 Submission postponed due to midterm.
3
  • 12.4: 33. Also find P(X = 5 | X < = 9) (Ans: 0.246) .
  • 12.4: 35, 41. For 41, 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: 47, 67, 69, 71, 73. For 73, 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) .
  • Find the expected value of the random variables whose p.m.f. appear in problem 12.4 - 9 (Ans: -0.1) and 12.4 - 21.
  • 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)
  • 12.4: 19. 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=2.2, 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: 21.
  • 12.4: 23. 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: 25, 27, 29.
  • 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 ~ HG(N=100, G=20, n=10). Y=2X+1.
      • (Ans: E(X)=2, V(X)=1.45, SD(X)=1.21, E(Y)=5, V(Y)=5.8, SD(Y)=2.42)
7/21/2011
4
  • 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).
  • 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)
  • 12.5: 5, 7, 9.
  • 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)
  • 12.5: 39, 41, 45, 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)
  • 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)
    		• 7/26/2011