Statistics 130AHomework # 2 Due:Tuesday, July 5, 2011
F. J. Samaniego
From the text: Section 1.3, # 10, 14
Section 2.1, # 14
Section 2.2, # 2, 12
Section 2.3, # 1, 6
Section 2.4, # 3, 6
Section 2.5, #4, 6, 12,17,22
Section 2.6, # 6, 8, 19, 21.
Also, solve the following problems:
1. A set of dominos is a collection of chips, each with two integers on it drawn from a set of n integers. Order plays no roll, as the chip [2, 3] occurs only once and can be played as either [2, 3] or [3, 2]. The standard domino set uses n = 7 integers and contains 28 pieces. Find the formula for the number of dominos in a set based on n integers.
2. There are 5 males and 5 females taking Statistics 177, “Little Known and Rarely Used Statistical Methods”. A team of three students is needed for a panel that will discuss the instructor’s research achievements.
a) How many different teams can be formed?
b) How many teams are there in which both genders are represented?
c) If John will only participate if Wendy participates and Veronica will participate only if her team is all female, how many different teams are possible?
3. A poker hand consists of five cards drawn at random without replacement from a standard 52-card deck. Find the probability of each of the following poker hands:
a) A straight flush (5 cards in a row of the same suit; Ace can be high or low)
b) Four of a kind (4 cards of one value, one of another)
c) Two pairs (2 cards of one value, 2 of another value, 1 of a third value)
d) One pair (2 cards of one value, I each from three other values)
4. Two balanced dice are rolled. Let X be the value facing up on the green die and Y be the value facing up on the red die. Let Z = max {X,Y}, that is, the larger of these two values (or their common value if X = Y). Obtain the probability distribution of Z, and evaluate EZ and Var (Z).
5. Moe, Larry and Curly, our favorite stooges, take turns tossing a fair coin, continuing to toss the coin, in that order, until one of them gets “tails”. The one who gets the first tails then gets to knock the other two guys’ heads together (yuk, yuk, yuk). Find the probability that Curly gets to do the head knocking. What’s the probability for Larry? And for Moe? (Hint: Curly wins on trials like HHT, HHHHHT, etc.)
6. Suppose that among all the widgets your company produces, the defective rate is .005. In the next batch of 1000 widgets produced, what is the probability that there are no more than three defectives? Find both the exact binomial probability and its Poisson approximation.
7. Suppose that X B(n,p), i.e. X has the Binomial distribution based on n trials of a dichotomous experiment with P(S) = p. Show that
E2X = (1 + p)n.
Note: It’s advisable for you to make a Xerox copy of your homework paper before turning it in, as it’s possible that the graded HWK # 2 papers will not be returned before Exam I on July 7.
Practice problems for Stat 130A Discussion Session, Thursday, June 30.
From the text: Section 2.1, # 18;Section 2.4, # 18; Section 2.5, #4; Section 2.6, # 18.
Also:
1. Suppose that two evenly matched teams (say team A and team B) make it to the baseball World Series. The series ends as soon as one of the teams has won four games. Thus, it can end as early as the 4th game (a “sweep”) or as late as the 7th game, with one team winning its fourth game after having lost three of the first six games. Calculate the probability that the series ends in 4 games, 5 games, 6 games or 7 games. What is the expected value of X, the number of games it takes to complete the series?
2.An elderly well-to-do developer has n properties and plans to divide them up in his will, giving them all away to his three children. If x1, x2 and x3 represent the number of properties to be inherited by heirs 1, 2 and 3, how many different distributions are possible? You may consider all properties of equal value. You need to find the number of possible triples (x1, x2, x3) for which xi is an integer ≥ 0 for each i and x1 + x2 + x3 = n. (Hint: Drawing a tree should help.)
3.The Educational Testing Service has published a study in which they assert that one of every six high school seniors will earn a college degree, and that one in every twenty will earn a graduate or professional degree. (a) If five high school seniors are selected at random, calculate the probability that exactly three of them will earn a college degree.
(b) If 30 high school seniors are selected at random, what’s the probability that at least two of them will earn a graduate or professional degree?
4. A fair coin is tossed until the fourth head is obtained. Find the probability that it takes at least seven tosses to obtain the fourth head.