- 8.1: Artemisia moves to a new house and she is " fifty-percent sure" tha...
- 8.2: Nefeli. a student in a probability class. takes a multiple-choice t...
- 8.3: The number of minutes between successive bus arrivals at Alvin's bu...
- 8.4: Students in a probability class take a multiple-choice test with 10...
- 8.5: Consider a variation of the biased coin problem in Example 8.4, and...
- 8.6: Professor May B. Hard, who has a tendency to give difficult problem...
- 8.7: We have two boxes, each containing three balls: one black and two w...
- 8.8: The probability of heads of a given coin is known to be either qo (...
- 8.9: Consider a Bayesian hypothesis testing problem involving m hypothes...
- 8.10: A police radar always overestimates the speed of incoming cars by a...
- 8.11: The number 8 of shopping carts in a store is uniformly distributed ...
- 8.12: Consider the multiple observation variant of Example 8.2: given tha...
- 8.13: (a) Let Y1, , Yn be independent identically distributed random vari...
- 8.14: Consider the random variables e and X in Example 8.11. Find the lin...
- 8.15: For the model in the shopping cart problem ( 11), derive and plot t...
- 8.16: The joint PDF of random variables X and e is of the form /x,e(x, O)...
- 8.17: Let e be a positive random variable, with known mean J..L and varia...
- 8.18: Swallowed Buft'on's needle. A doctor is treating a patient who has ...
- 8.19: Consider a photodetector in an optical communications system that c...
- 8.20: Estimation with spherically invariant PDFs. Let 8 and X be continuo...
- 8.21: Linear LMS estimation based on two observations. Consider three ran...
- 8.22: Linear LMS estimation based on multiple observations. Let 8 be a ra...
- 8.23: Properties of LMS estimation. Let e and X be two random variables w...
- 8.24: Properties of linear LMS estimation based on multiple observations....
Solutions for Chapter 8: Bayesian Statistical Inference
Full solutions for Introduction to Probability, | 2nd Edition
Additivity property of x 2
If two independent random variables X1 and X2 are distributed as chi-square with v1 and v2 degrees of freedom, respectively, Y = + X X 1 2 is a chi-square random variable with u = + v v 1 2 degrees of freedom. This generalizes to any number of independent chi-square random variables.
In a fractional factorial experiment when certain factor effects cannot be estimated uniquely, they are said to be aliased.
In statistical hypothesis testing, this is a hypothesis other than the one that is being tested. The alternative hypothesis contains feasible conditions, whereas the null hypothesis speciies conditions that are under test
Binomial random variable
A discrete random variable that equals the number of successes in a ixed number of Bernoulli trials.
The joint probability distribution of two random variables.
Bivariate normal distribution
The joint distribution of two normal random variables
Data consisting of counts or observations that can be classiied into categories. The categories may be descriptive.
Chi-square (or chi-squared) 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.
An experiment in which the treatments (experimental conditions) that are to be studied are included in the experiment. The data from the experiment are used to evaluate the treatments.
Completely randomized design (or experiment)
A type of experimental design in which the treatments or design factors are assigned to the experimental units in a random manner. In designed experiments, a completely randomized design results from running all of the treatment combinations in random order.
Conditional probability mass function
The probability mass function of the conditional probability distribution of a discrete random variable.
A tabular arrangement expressing the assignment of members of a data set according to two or more categories or classiication criteria
A method to derive the probability density function of the sum of two independent random variables from an integral (or sum) of probability density (or mass) functions.
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).
Formulas used to determine the number of elements in sample spaces and events.
Discrete random variable
A random variable with a inite (or countably ininite) range.
Discrete uniform random variable
A discrete random variable with a inite range and constant probability mass function.
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.
A function used in the probability density function of a gamma random variable that can be considered to extend factorials
Another name for the normal distribution, based on the strong connection of Karl F. Gauss to the normal distribution; often used in physics and electrical engineering applications