 7.1: Alice and Bob arrange to meet for lunch on a certain day at noon. H...
 7.2: Alice, Bob, and Carl arrange to meet for lunch on a certain day. Th...
 7.3: One of two doctors, Dr. Hibbert and Dr. Nick, is called upon to per...
 7.4: A fair coin is flipped twice. Let X be the number of Heads in the t...
 7.5: A fair die is rolled, and then a coin with probability p of Heads i...
 7.6: A committee of size k is chosen from a group of n women and m men. ...
 7.7: A stick of length L (a positive constant) is broken at a uniformly ...
 7.8: (a) Five cards are randomly chosen from a standard deck, one at a t...
 7.9: Let X and Y be i.i.d. Geom(p), and N = X + Y . (a) Find the joint P...
 7.10: Let X and Y be i.i.d. Expo(), and T = X + Y . (a) Find the conditio...
 7.11: Let X, Y, Z be r.v.s such that X N (0, 1) and conditional on X = x,...
 7.12: Let X Expo(), and let c be a positive constant. (a) If you remember...
 7.13: Let X and Y be i.i.d. Expo(). Find the conditional distribution of ...
 7.14: (a) A stick is broken into three pieces by picking two points indep...
 7.15: Let X and Y be continuous r.v.s., with joint CDF F(x, y). Show that...
 7.16: Let X and Y have joint PDF fX,Y (x, y) = x + y, for 0 <x< 1 and 0 <...
 7.17: Let X and Y have joint PDF fX,Y (x, y) = cxy, for 0 0 <x<y< 1. (a) ...
 7.18: Let (X, Y ) be a uniformly random point in the triangle in the plan...
 7.19: A random point (X, Y, Z) is chosen uniformly in the ball B = {(x, y...
 7.20: Let U1, U2, U3 be i.i.d. Unif(0, 1), and let L = min(U1, U2, U3), M...
 7.21: Find the probability that the quadratic polynomial Ax2 +Bx+ 1, wher...
 7.22: Let X and Y each have support (0, 1) marginally, and suppose that t...
 7.23: The unit ball in Rn is {(x1,...,xn) : x2 1 + + x2 n 1}, the ball of...
 7.24: Two students, A and B, are working independently on homework (not n...
 7.25: Two companies, Company 1 and Company 2, have just been founded. Sto...
 7.26: The bus company from Blissville decides to start service in Blotchv...
 7.27: A longevity study is being conducted on n married hobbit couples. L...
 7.28: There are n stores in a shopping center, labeled from 1 to n. Let X...
 7.29: Let X and Y be i.i.d. Geom(p), L = min(X, Y ), and M = max(X, Y ). ...
 7.30: Let X, Y have the joint CDF F(x, y)=1 e x e y + e (x+y+xy) , for x ...
 7.31: Let X and Y be i.i.d. Unif(0, 1). Find the standard deviation of th...
 7.32: Let X, Y be i.i.d. Expo(). Find EX Y  in two dierent ways: (a) us...
 7.33: Alice walks into a post oce with 2 clerks. Both clerks are in the m...
 7.34: Let (X, Y ) be a uniformly random point in the triangle in the plan...
 7.35: A random point is chosen uniformly in the unit disk {(x, y) : x2 + ...
 7.36: Let X and Y be discrete r.v.s. (a) Use 2D LOTUS (without assuming l...
 7.37: Let X and Y be i.i.d. continuous random variables with PDF f, mean ...
 7.38: Let X and Y be r.v.s. Is it correct to say max(X, Y ) + min(X, Y ) ...
 7.39: Two fair sixsided dice are rolled (one green and one orange), with...
 7.40: Let X and Y be i.i.d. Unif(0, 1). (a) Compute the covariance of X +...
 7.41: Let X and Y be standardized r.v.s (i.e., marginally they each have ...
 7.42: Let X be the number of distinct birthdays in a group of 110 people ...
 7.43: Let X and Y be Bernoulli r.v.s, possibly with dierent parameters. S...
 7.44: Find the variance of the number of toys needed until you have a com...
 7.45: A random triangle is formed in some way, such that the angles are i...
 7.46: Each of n 2 people puts his or her name on a slip of paper (no two ...
 7.47: Athletes compete one at a time at the high jump. Let Xj be how high...
 7.48: A chicken lays a Pois() number N of eggs. Each egg hatches a chick ...
 7.49: Let X1,...,Xn be random variables such that Corr(Xi, Xj ) = for all...
 7.50: Let X and Y be independent r.v.s. Show that Var(XY ) = Var(X)Var(Y ...
 7.51: Stat 110 shirts come in 3 sizes: small, medium, and large. There ar...
 7.52: A drunken man wanders around randomly in a large space. At each ste...
 7.53: A scientist makes two measurements, considered to be independent st...
 7.54: Let U Unif(1, 1) and V = 2U 1. (a) Find the distribution of V (gi...
 7.55: Consider the following method for creating a bivariate Poisson (a j...
 7.56: You are playing an exciting game of Battleship. Your opponent secre...
 7.57: This problem explores a visual interpretation of covariance. Data a...
 7.58: A statistician is trying to estimate an unknown parameter based on ...
 7.59: A Pois() number of people vote in a certain election. Each voter vo...
 7.60: A traveler gets lost N Pois() times on a long journey. When lost, t...
 7.61: The number of people who visit the Leftorium store in a day is Pois...
 7.62: A chicken lays n eggs. Each egg independently does or doesnt hatch,...
 7.63: There will be X Pois() courses oered at a certain school next year....
 7.64: Let (X1,...,Xk) be Multinomial with parameters n and (p1,...,pk). U...
 7.65: Consider the birthdays of 100 people. Assume peoples birthdays are ...
 7.66: A certain course has a freshmen, b sophomores, c juniors, and d sen...
 7.67: A group of n 2 people decide to play an exciting game of RockPaper...
 7.68: Emails arrive in an inbox according to a Poisson process with rate ...
 7.69: Let X be the number of statistics majors in a certain college in th...
 7.70: In humans (and many other organisms), genes come in pairs. Consider...
 7.71: Let (X, Y ) be Bivariate Normal, with X and Y marginally N (0, 1) a...
 7.72: Let the joint PDF of X and Y be fX,Y (x, y) = c exp x2 2 y2 2 for a...
 7.73: Let the joint PDF of X and Y be fX,Y (x, y) = c exp x2 2 y2 2 for x...
 7.74: Let X, Y, Z be i.i.d. N (0, 1). Find the joint MGF of (X + 2Y, 3X +...
 7.75: Let X and Y be i.i.d. N (0, 1), and let S be a random sign (1 or 1,...
 7.76: Let (X, Y ) be Bivariate Normal with X N (0, 2 1) and Y N (0, 2 2) ...
 7.77: A mother and a father have 6 children. The 8 heights in the family ...
 7.78: Cars pass by a certain point on a road according to a Poisson proce...
 7.79: In a U.S. election, there will be V Pois() registered voters. Suppo...
 7.80: A certain college has m freshmen, m sophomores, m juniors, and m se...
 7.81: Let X Expo() and let Y be a random variable, discrete or continuous...
 7.82: To test for a certain disease, the level of a certain substance in ...
 7.83: Let J be Discrete Uniform on {1, 2,...,n}. (a) Find E(J) and Var(J)...
 7.84: A network consists of n nodes, each pair of which may or may not ha...
 7.85: Shakespeare wrote a total of 884647 words in his known works. Of co...
Solutions for Chapter 7: Joint Distributions
Full solutions for Introduction to Probability  1st Edition
ISBN: 9781466575578
Solutions for Chapter 7: Joint Distributions
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2 k factorial experiment.
A full factorial experiment with k factors and all factors tested at only two levels (settings) each.

`error (or `risk)
In hypothesis testing, an error incurred by rejecting a null hypothesis when it is actually true (also called a type I error).

Bayes’ estimator
An estimator for a parameter obtained from a Bayesian method that uses a prior distribution for the parameter along with the conditional distribution of the data given the parameter to obtain the posterior distribution of the parameter. The estimator is obtained from the posterior distribution.

Box plot (or box and whisker plot)
A graphical display of data in which the box contains the middle 50% of the data (the interquartile range) with the median dividing it, and the whiskers extend to the smallest and largest values (or some deined lower and upper limits).

C chart
An attribute control chart that plots the total number of defects per unit in a subgroup. Similar to a defectsperunit or U chart.

Center line
A horizontal line on a control chart at the value that estimates the mean of the statistic plotted on the chart. See Control chart.

Chisquare (or chisquared) random variable
A continuous random variable that results from the sum of squares of independent standard normal random variables. It is a special case of a gamma random variable.

Confounding
When a factorial experiment is run in blocks and the blocks are too small to contain a complete replicate of the experiment, one can run a fraction of the replicate in each block, but this results in losing information on some effects. These effects are linked with or confounded with the blocks. In general, when two factors are varied such that their individual effects cannot be determined separately, their effects are said to be confounded.

Conidence coeficient
The probability 1?a associated with a conidence interval expressing the probability that the stated interval will contain the true parameter value.

Cumulative distribution function
For a random variable X, the function of X deined as PX x ( ) ? that is used to specify the probability distribution.

Decision interval
A parameter in a tabular CUSUM algorithm that is determined from a tradeoff between false alarms and the detection of assignable causes.

Deming
W. Edwards Deming (1900–1993) was a leader in the use of statistical quality control.

Enumerative study
A study in which a sample from a population is used to make inference to the population. See Analytic study

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.

Estimator (or point estimator)
A procedure for producing an estimate of a parameter of interest. An estimator is usually a function of only sample data values, and when these data values are available, it results in an estimate of the parameter of interest.

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.

Fixed factor (or fixed effect).
In analysis of variance, a factor or effect is considered ixed if all the levels of interest for that factor are included in the experiment. Conclusions are then valid about this set of levels only, although when the factor is quantitative, it is customary to it a model to the data for interpolating between these levels.

Gamma function
A function used in the probability density function of a gamma random variable that can be considered to extend factorials

Generating function
A function that is used to determine properties of the probability distribution of a random variable. See Momentgenerating function

Generator
Effects in a fractional factorial experiment that are used to construct the experimental tests used in the experiment. The generators also deine the aliases.