- 11.2.1: In 1-8, the general solution of the linear system X' = AX is given.
- 11.2.2: In 1-8, the general solution of the linear system X' = AX is given.
- 11.2.3: In 1-8, the general solution of the linear system X' = AX is given.
- 11.2.4: In 1-8, the general solution of the linear system X' = AX is given....
- 11.2.5: In 1-8, the general solution of the linear system X' = AX is given....
- 11.2.6: In 1-8, the general solution of the linear system X' = AX is given....
- 11.2.7: In 1-8, the general solution of the linear system X' = AX is given....
- 11.2.8: In 1-8, the general solution of the linear system X' = AX is given....
- 11.2.9: x' = -5x + 3y y' = 2x + ?y
- 11.2.10: x' = -5x + 3y y' = 2x- ?y
- 11.2.11: x' = -5x + 3y y' = -2x + Sy
- 11.2.12: x' = -5x + 3y y' = -?x + 4y
- 11.2.13: x' = - x + i y y' = -x - b
- 11.2.14: x' = x + i y y' = -x + b
- 11.2.15: x' = 0.02x - O.lly y' = O.lOx - 0.05y
- 11.2.16: x' = 0.03x + O.Oly y' = -0.0lx + 0.05y
- 11.2.17: Determine conditions on the real constant so that (0, 0) is a cente...
- 11.2.18: Determine a condition on the real constant so that (0, 0) is a stab...
- 11.2.19: Show that (0, 0) is always an unstable critical point of the linear...
- 11.2.20: Let X = X(t) be the response of the linear dynamical system x ' =ax...
- 11.2.21: Showthat the nonhomogeneouslinear systemX' =AX+ F has a unique crit...
- 11.2.22: In Example 4(b) show that (0, 0) is a stable node when be< 1.
- 11.2.23: x' = 2x + 3y - 6 y' = -x- 2y + 5
- 11.2.24: x' = -5x + 9y + 13 y' = -x - lly - 23
- 11.2.25: x' = O.lx - 0.2y + 0.35 y' = O.lx + O.ly - 0.25
- 11.2.26: x' = 3x - 2y - 1 y' = 5x - 3y - 2
Solutions for Chapter 11.2: Stability of Linear Systems
Full solutions for Advanced Engineering Mathematics | 5th Edition
a-error (or a-risk)
In hypothesis testing, an error incurred by failing to reject a null hypothesis when it is actually false (also called a type II error).
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.
Adjusted R 2
A variation of the R 2 statistic that compensates for the number of parameters in a regression model. Essentially, the adjustment is a penalty for increasing the number of parameters in the model. Alias. In a fractional factorial experiment when certain factor effects cannot be estimated uniquely, they are said to be aliased.
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
Asymptotic relative eficiency (ARE)
Used to compare hypothesis tests. The ARE of one test relative to another is the limiting ratio of the sample sizes necessary to obtain identical error probabilities for the two procedures.
A chart used to organize the various potential causes of a problem. Also called a ishbone diagram.
Conditional probability density function
The probability density function of the conditional probability distribution of a continuous random variable.
An estimator that converges in probability to the true value of the estimated parameter as the sample size increases.
A tabular arrangement expressing the assignment of members of a data set according to two or more categories or classiication criteria
A square matrix that contains the variances and covariances among a set of random variables, say, X1 , X X 2 k , , … . The main diagonal elements of the matrix are the variances of the random variables and the off-diagonal elements are the covariances between Xi and Xj . Also called the variance-covariance matrix. When the random variables are standardized to have unit variances, the covariance matrix becomes the correlation matrix.
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.
Cumulative distribution function
For a random variable X, the function of X deined as PX x ( ) ? that is used to specify the probability distribution.
Degrees of freedom.
The number of independent comparisons that can be made among the elements of a sample. The term is analogous to the number of degrees of freedom for an object in a dynamic system, which is the number of independent coordinates required to determine the motion of the object.
An experiment in which the tests are planned in advance and the plans usually incorporate statistical models. See Experiment
A probability distribution for a discrete random variable
Discrete random variable
A random variable with a inite (or countably ininite) range.
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.
A subset of a sample space.
The expected value of a random variable X is its long-term average or mean value. In the continuous case, the expected value of X is E X xf x dx ( ) = ?? ( ) ? ? where f ( ) x is the density function of the random variable X.
The distribution of the random variable deined as the ratio of two independent chi-square random variables, each divided by its number of degrees of freedom.