Hints

• Assignment 7

  Number 4.3.9:

  The book's answer is wrong for part (b). You should get 0.123. The sum in part (a) is .1542, which is the exact answer. Normal approximations to binomial probabilities are best when np = .5. Here np=.7.

• Review Problems For Midterm II

  3.9.7:

  Label the 48 non-aces 1 through 48. Let Xi = 1 if the ith card (using our labeling scheme) is on top of the first Ace, 0 else. The four aces divide the deck into five segments, and by symmetry, a given (non-ace) card is just as likely to be in one segment as any other. Thus, the P(Xi = 1) = 1/5. (Xi=1 if the card appears in the first segment) Now, let X = number of cards on top of the first Ace. Then X = X1 + X2 +... + X48. So E(X) = 48*1/5 = 9.6.

  3.9.9:

  Let Xi = i if the ith chip is chosen, 0 else. Since we are selecting r chips, the probability that the ith chip is chosen is r/n. So, let X = sum of the r chips chosen = X1 + X2 + ... Xn. So E(X) = (r/n)*(1 + 2 + .... + n) = (r/n)*(n(n+1)/2) = r(n+1)/2. To see why 1 + 2 + ... + n = n(n+1)/2, imagine a matrix in which the first column is a 1 followed by n-1 0's, the second column has two 1's followed by n-2 zeros, and the nth column is all 1s. Then what we want to do is count the number of 1's in the matrix. The diagonal has n 1's along it. The total number of entries is n^2 (its an n by n matrix), and the number of entries above the diagonal is s. The total number of entries in the matrix is the number of entries on the diagonal plus the number above plus the number below, hence: n^2 = 2s + n. This means that s = n(n-1)/2. We want the number of entries above the diagonal plus the number on the diagonal, or n(n-1)/2 + n = n(n+1)/2.

  ---

  Assignment 5

  Number 3.9.2: The question is asking for constraints on the ai's. In other words, for which ai's will the equality be true?

  ---

  Assignment 2

  2.5.2

  f(t) is a density. What properties do densities have? k must be chosen so that these properties hold.

  2.5.7

  The same hint (practically) applies. The function to be integrated is almost a density. The function above, f(x), is a density.

  
  

  ---

  Home