 2.1: The MIT soccer team has 2 games scheduled for one weekend. It has a...
 2.2: You go to a party with 500 guests. What is the probability that exa...
 2.3: Fischer and Spassky play a chess match in which the first player to...
 2.4: An internet service provider uses 50 modems to serve the needs of 1...
 2.5: A packet communication system consists of a buffer that stores pack...
 2.6: The Celtics and the Lakers are set to play a playoff series of n ba...
 2.7: You just rented a large house and the realtor gave you 5 keys, one ...
 2.8: Recursive computation of the binomial PMF. Let X be a binomial rand...
 2.9: Form of the binomial PMF. Consider a binomial random variable X wit...
 2.10: Form of the Poisson PMF. Let X be a Poisson random variable with pa...
 2.11: The matchbox problem  inspired by Banach's smoking habits. A smoke...
 2.12: Justification of the Poisson approximation property. Consider the P...
 2.13: A family has 5 natural children and has adopted 2 girls. Each natur...
 2.14: Let X be a random variable that takes values from to 9 with equal p...
 2.15: Let K be a random variable that takes, with equal probability 1/(2n...
 2.16: Let X be a random variable with PMF ( ) { X 2 la, px x = 0, (a) Fin...
 2.17: A city's temperature is modeled as a random variable with mean and ...
 2.18: Let a and b be positive integers with a ::; b, and let X be a rando...
 2.19: A prize is randomly placed in one of ten boxes, numbered from 1 to ...
 2.20: As an advertising campaign, a chocolate factory places golden ticke...
 2.21: St. Petersburg paradox. You toss independently a fair coin and you ...
 2.22: Two coins are simultaneously tossed until one of them comes up a he...
 2.23: (a) A fair coin is tossed repeatedly and independently until two co...
 2.24: A stock market trader buys 100 shares of stock A and 200 shares of ...
 2.25: A class of n students takes a test consisting of m questions. Suppo...
 2.26: PMF of the minimum of several random variables. On a given day. you...
 2.27: The multinomial distribution. A die with r faces, numbered 1, .... ...
 2.28: The quiz problem. Consider a quiz contest where a person is given a...
 2.29: The inclusionexclusion formula. Let AI , A2, . .. , An be events. ...
 2.30: Alvin's database of friends contains n entries, but due to a softwa...
 2.31: Consider four independent rolls of a 6sided die. Let X be the numb...
 2.32: D. Bernoulli's problem of joint lives. Consider 2m persons forming ...
 2.33: A coin that has probability of heads equal to p is tossed successiv...
 2.34: A spider and a fly move along a straight line. At each second, the ...
 2.35: Verify the expected value rule E[g(X, Y)] = L Lg(x,y)pX,y(x,y), x y...
 2.36: The multiplication rule for conditional PMFs. Let X, Y, and Z be ra...
 2.37: Splitting a Poisson random variable. A transmitter sends out either...
 2.38: Alice passes through four traffic lights on her way to work, and ea...
 2.39: Each morning, Hungry Harry eats some eggs. On any given morning, th...
 2.40: A particular professor is known for his arbitrary grading policies....
 2.41: You drive to work 5 days a week for a full year (50 weeks), and wit...
 2.42: Computational problem. Here is a probabilistic method for computing...
 2.43: Suppose that X and Y are independent, identically distributed, geom...
 2.44: Let X and Y be two random variables with given joint PMF, and let 9...
 2.45: Variability extremes. Let Xl , . . . , X n be independent random va...
 2.46: Entropy and uncertainty. Consider a random variable X that can take...
Solutions for Chapter 2: Discrete Random Variables
Full solutions for Introduction to Probability,  2nd Edition
ISBN: 9781886529236
Solutions for Chapter 2: Discrete Random Variables
Get Full SolutionsChapter 2: Discrete Random Variables includes 46 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Probability,, edition: 2. Introduction to Probability, was written by and is associated to the ISBN: 9781886529236. Since 46 problems in chapter 2: Discrete Random Variables have been answered, more than 7418 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Addition rule
A formula used to determine the probability of the union of two (or more) events from the probabilities of the events and their intersection(s).

Arithmetic mean
The arithmetic mean of a set of numbers x1 , x2 ,…, xn is their sum divided by the number of observations, or ( / )1 1 n xi t n ? = . The arithmetic mean is usually denoted by x , and is often called the average

Attribute control chart
Any control chart for a discrete random variable. See Variables control chart.

Axioms of probability
A set of rules that probabilities deined on a sample space must follow. See Probability

Bimodal distribution.
A distribution with two modes

Binomial random variable
A discrete random variable that equals the number of successes in a ixed number of Bernoulli trials.

Bivariate normal distribution
The joint distribution of two normal random variables

Causeandeffect diagram
A chart used to organize the various potential causes of a problem. Also called a ishbone diagram.

Combination.
A subset selected without replacement from a set used to determine the number of outcomes in events and sample spaces.

Continuous distribution
A probability distribution for a continuous random variable.

Correlation coeficient
A dimensionless measure of the linear association between two variables, usually lying in the interval from ?1 to +1, with zero indicating the absence of correlation (but not necessarily the independence of the two variables).

Critical value(s)
The value of a statistic corresponding to a stated signiicance level as determined from the sampling distribution. For example, if PZ z PZ ( )( .) . ? =? = 0 025 . 1 96 0 025, then z0 025 . = 1 9. 6 is the critical value of z at the 0.025 level of signiicance. Crossed factors. Another name for factors that are arranged in a factorial experiment.

Density function
Another name for a probability density function

Eficiency
A concept in parameter estimation that uses the variances of different estimators; essentially, an estimator is more eficient than another estimator if it has smaller variance. When estimators are biased, the concept requires modiication.

Error propagation
An analysis of how the variance of the random variable that represents that output of a system depends on the variances of the inputs. A formula exists when the output is a linear function of the inputs and the formula is simpliied if the inputs are assumed to be independent.

Error sum of squares
In analysis of variance, this is the portion of total variability that is due to the random component in the data. It is usually based on replication of observations at certain treatment combinations in the experiment. It is sometimes called the residual sum of squares, although this is really a better term to use only when the sum of squares is based on the remnants of a modelitting process and not on replication.

Event
A subset of a sample space.

Experiment
A series of tests in which changes are made to the system under study

Extra sum of squares method
A method used in regression analysis to conduct a hypothesis test for the additional contribution of one or more variables to a model.

Ftest
Any test of signiicance involving the F distribution. The most common Ftests are (1) testing hypotheses about the variances or standard deviations of two independent normal distributions, (2) testing hypotheses about treatment means or variance components in the analysis of variance, and (3) testing signiicance of regression or tests on subsets of parameters in a regression model.