Homework 1


due in class, Tuesday October 5



  1. A coin is to be tossed as many times as necesary to turn up one head. Thus the elements, c, of the sample space S are H, TH, TTH, TTTH, and so forth. Let the probability set function P assign to these elements the respective probabilities 1/2, 1/4, 1/8, 1/16, and so forth. Show that P(S) = 1. Let C_1 = {c: c is H, TH, TTH, TTTH, or TTTTH}. Compute P(C_1). Let C_2 = {c: c is TTTTH or TTTTTH}. Compute P(C_2), P(C_1 and C_2), and P(C_1 or C_2).
  2. If C_1 and C_2 are subsets of the sample space S, show that P(C_1 and C_2) <= P(C_1) <= P(C_1 or C_2) <= P(C_1) + P(C_2) . (Note that C_1 and C_2 are unspecified sets, this question is separate from the previous question.)
  3. There are 5 red chips and 3 blue chips in a bowl. The red chips are numbered 1,2,3,4,5 respectively, and the blue chips are numbered 1,2,3 respectively. If 2 chips are drawn at random and without replacement, find the probability that these chips have either the same number or the same color.
  4. Bowl I contains 6 red chips and 4 blue chips. Five of these 10 chips are selected at random and without replacement and put in bowl II, which was originally empty. One chip is then drawn at random from bowl II. Given that this chip is blue, find the conditional probability that 2 red chips and 3 blue chips are transferred from bowl I to bowl II.
  5. Suppose five cards are selected at random and without replacement from an ordinary deck of playing cards. Find the probability mass function of X, the number of hearts in the five cards. Compute P(X <= 1).
  6. Let f(x) = 1/(x^2), 1 < x < infinity, zero elsewhere, be the p.d.f. of X. Find P(-2 < x < 3).
  7. A mode of a distribution of one random variable X is a value of x that maximizes the p.d.f. f(x). For X of the continuous type, f(x) must be continuous. If there is only one such x, it is called the mode of the distribution. Find the mode of each of the following distributions:
    (a) f(x) = (1/2)^x, x=1,2,3,..., zero elsewhere
    (b) f(x) = 12(x^2)(1-x), 0 < x < 1, zero elsewhere
    (c) f(x) = (1/2)(x^2) exp(-x), 0 < x < infinity, zero elsewhere
  8. Let X have p.d.f. f(x) = 2 * x * exp(-x^2), 0 < x < infinity, zero elsewhere. Find the p.d.f. of Y = X^2.
  9. Let X have p.d.f. f(x) = 3(x^2), 0 < x < 1, zero elsewhere. Consider a random rectangle whose sides are X and (1-X). Determine the expected value of the area of the rectangle.
  10. Let f(x) = (1/2)^x, x = 1,2,3,..., zero elsewhere, be the p.d.f. of the random variable X. Find the m.g.f., the mean, and the variance of X.