{"id":19,"date":"2023-11-27T14:11:08","date_gmt":"2023-11-27T19:11:08","guid":{"rendered":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/?post_type=front-matter&p=19"},"modified":"2024-01-05T09:53:56","modified_gmt":"2024-01-05T14:53:56","slug":"terms-and-formulas","status":"publish","type":"front-matter","link":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/front-matter\/terms-and-formulas\/","title":{"raw":"Terms and Formulas","rendered":"Terms and Formulas"},"content":{"raw":"
[Latexpage]<\/div>\r\n\r\n\r\n\r\n
Term<\/strong><\/td>\r\nFormula<\/strong><\/td>\r\nExplanation<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Ratio<\/td>\r\na <\/em>: b<\/em> <\/span><\/td>\r\n<\/td>\r\n<\/tr>\r\n
Proportion<\/td>\r\n$\\frac{a}{b}=\\frac{c}{d}$<\/span><\/td>\r\nAn equation with a ratio (or rate) on two sides, in which the two ratios are equal.<\/td>\r\n<\/tr>\r\n
Cross-product rule<\/td>\r\n$\\frac{a}{b} = \\frac{c}{d} \\quad \\text{equals} \\quad a \\times d = c \\times b$<\/span><\/td>\r\nMultiplying along two diagonals and solving for the unknown.<\/td>\r\n<\/tr>\r\n
Percent proportion method<\/td>\r\n$\\frac{\\text{Part (\"is\")}}{\\text{Whole (\"of\")}} = \\frac{\\%}{100}$<\/span><\/td>\r\n<\/td>\r\n<\/tr>\r\n
Percent increase<\/td>\r\n= $\\frac{New\\;value-Original\\;value}{Original\\;value}$<\/span><\/td>\r\n<\/td>\r\n<\/tr>\r\n
Percent decrease<\/td>\r\n= $\\frac{Original\\;value-New\\;value}{Original\\;value}$<\/span><\/td>\r\n<\/td>\r\n<\/tr>\r\n
Linear equation: slope-intercept form<\/td>\r\ny = m<\/em>x + b<\/em><\/td>\r\nm\u00a0<\/em>is the slope and b <\/em>is the constant.<\/td>\r\n<\/tr>\r\n
Slope of a linear equation<\/td>\r\n$m = \\frac{y_2 - y_1}{x_2 - x_1}$<\/span><\/td>\r\n$(x_1, y_1) \\text{ and } (x_2, y_2) \\text{ are two points.}$<\/em><\/td>\r\n<\/tr>\r\n
Linear equation: standard form<\/td>\r\nAx + By = C<\/span><\/td>\r\n\u00a0<\/em><\/td>\r\n<\/tr>\r\n
Slope of a linear equation in standard form<\/td>\r\n$m = -\\frac{A}{B}$<\/span><\/td>\r\n\u00a0<\/em><\/td>\r\n<\/tr>\r\n
Linear equation: Point-slope formula<\/td>\r\n$y-y_1=m(x-x_1)$<\/span><\/td>\r\n$(x_1, y_1) \\text{ is a point.}$<\/span><\/em><\/td>\r\n<\/tr>\r\n
Average rate of change<\/td>\r\n$\\frac{Change\\;of\\;Output}{Change\\;of\\;Input}=\\frac{\\Delta y}{\\Delta x}=\\frac{y_2-y_1}{ x_2-x_1}$<\/span><\/td>\r\nThe average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.<\/td>\r\n<\/tr>\r\n
Quadratic function: standard form<\/td>\r\n$f(x)=ax^2+bx+c$<\/span><\/td>\r\n\u00a0<\/span><\/em><\/td>\r\n<\/tr>\r\n
Quadratic function: vertex form<\/td>\r\n$f(x)=a(x-h)^2+k$<\/span><\/td>\r\n$(h, k) \\text{ is the vertex point.}$<\/span><\/em><\/td>\r\n<\/tr>\r\n
Vertex of a quadratic function<\/td>\r\n\r\n

$h=-\\frac{b}{2a}$,\u00a0 \u00a0$k=f(h)=f(\\frac{-b}{2a})$<\/p>\r\n<\/td>\r\n

\u00a0<\/span><\/em><\/td>\r\n<\/tr>\r\n
Quadratic formula<\/td>\r\n$x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}}$<\/td>\r\nIt gives the horizontal intercepts.<\/td>\r\n<\/tr>\r\n
Exponential growth or decay function<\/td>\r\n$f(x)=a(1+r)^x$\u00a0 \u00a0or\u00a0 \u00a0$f(x)=ab^x$\u00a0 \u00a0where $b=1+r$<\/td>\r\na<\/em> is the initial or starting value, r<\/em> is the percent growth or decay rate, and b<\/em> is the growth factor.\r\n\r\n$\\text{Note: if we have a decay rate, } 1 + r \\text{ should be changed to } 1 - r \\text{.}$<\/span><\/td>\r\n<\/tr>\r\n
Compound Interest Formula<\/td>\r\n$A(t)=a(1+\\frac{r}{k})^{kt}$<\/td>\r\nr<\/em> is the annual percentage rate (APR), also called the nominal rate, and k<\/em> is the number of compounding periods in one year<\/td>\r\n<\/tr>\r\n
Continuous Growth Formula<\/td>\r\nf(x) = $ae^{rx}$<\/span><\/td>\r\n$\\text{Note: if we have a decay rate, } r \\text{ should be changed to } - r \\text{.}$<\/span><\/em><\/td>\r\n<\/tr>\r\n
Conversion to log form<\/td>\r\nba<\/sup><\/em> = c<\/em> is equivalent to the statement logb<\/em><\/sub>(c<\/em>) = a<\/em> .<\/td>\r\n\u00a0<\/span><\/em><\/td>\r\n<\/tr>\r\n
Properties of Logs: Inverse Properties<\/td>\r\nlogb<\/em><\/sub>(bx<\/sup><\/em>) = x<\/em>\r\n\r\nb<\/em>logb<\/sub>x<\/em><\/sup> = x<\/em><\/td>\r\n\u00a0<\/span><\/em><\/td>\r\n<\/tr>\r\n
Properties of Logs: Exponent Property<\/td>\r\n\r\n

logb<\/em><\/sub>(Ar<\/sup><\/em>) = r<\/em>logb<\/em><\/sub>(A<\/em>)<\/p>\r\n<\/td>\r\n

\u00a0<\/span><\/em><\/td>\r\n<\/tr>\r\n
Properties of Logs: Change of Base<\/td>\r\n\r\n

$\\log_{b}(A)=\\frac{\\log_{c}(A)}{\\log_{c}(b)}$<\/p>\r\n<\/td>\r\n

\u00a0<\/span><\/em><\/td>\r\n<\/tr>\r\n
Sum of Logs Property<\/td>\r\n\r\n

$\\log_{b}(A)+\\log_{b}(C)=\\log_b(AC)$<\/p>\r\n<\/td>\r\n

\u00a0<\/span><\/em><\/td>\r\n<\/tr>\r\n
Difference of Logs Property<\/td>\r\n\r\n

$\\log_{b}(A)-log_{b}(C)=\\log_{b}(\\frac{A}{C})$<\/p>\r\n<\/td>\r\n

\u00a0<\/span><\/em><\/td>\r\n<\/tr>\r\n
Half-Life (Based on standard exponential function)<\/td>\r\n\r\n

$\\frac{1}{2}=b^t$<\/p>\r\n<\/td>\r\n

\u00a0<\/span><\/em><\/td>\r\n<\/tr>\r\n
Half-Life (Based on continuous change function)<\/td>\r\n\r\n

$\\frac{1}{2}=e^{rt}$<\/p>\r\n<\/td>\r\n

\u00a0<\/span><\/em><\/td>\r\n<\/tr>\r\n
Doubling Time (Based on standard exponential function)<\/td>\r\n\r\n

2 = bt<\/sup><\/em><\/p>\r\n<\/td>\r\n

\u00a0<\/span><\/em><\/td>\r\n<\/tr>\r\n
Half-Life (Based on continuous change function)<\/td>\r\n\r\n

$2=e^{rt}$<\/p>\r\n<\/td>\r\n

\u00a0<\/span><\/em><\/td>\r\n<\/tr>\r\n
Subset<\/td>\r\n\r\n

$A\\subseteq B$<\/p>\r\n<\/td>\r\n

set A is a subset of a set B if every member of A is also a member of B.<\/td>\r\n<\/tr>\r\n
Union of two sets<\/td>\r\n\r\n

$A\\cup B$<\/p>\r\n<\/td>\r\n

The set of all elements that are either in A or in B, or in both.<\/td>\r\n<\/tr>\r\n
Intersection of two sets<\/td>\r\n\r\n

$A\\cap B$<\/p>\r\n<\/td>\r\n

The set of all elements that are common to both sets A and B.<\/td>\r\n<\/tr>\r\n
Complement of a set<\/td>\r\n\r\n

$A^c$<\/p>\r\n<\/td>\r\n

The set consists of elements in the universal set U that are not in A.<\/td>\r\n<\/tr>\r\n
Permutations<\/td>\r\n\r\n

n<\/em>Pr<\/em> = $\\frac{n!}{(n-r)!}$<\/p>\r\n<\/td>\r\n

The Number of Permutations of n<\/em> Objects Taken r<\/em> at a Time.<\/td>\r\n<\/tr>\r\n
Circular Permutations<\/td>\r\n\r\n

(n \u2212 1)!<\/em><\/p>\r\n<\/td>\r\n

The number of permutations of n<\/em> elements in a circle.<\/td>\r\n<\/tr>\r\n
Permutations with Similar Elements<\/td>\r\n\r\n

$\\frac{n!}{r_1!r_2!...r_k!}$<\/em><\/p>\r\n<\/td>\r\n

The number of permutations of n<\/em> elements taken n<\/em> at a time, with r<\/em>1<\/sub> elements of one kind, r<\/em>2<\/sub> elements of another kind, and so on.<\/td>\r\n<\/tr>\r\n
Combinations<\/td>\r\nn<\/em>Cr<\/em> = $\\frac{n!}{(n-r)!r!}$<\/td>\r\nThe Number of Combinations of n<\/em> Objects Taken r<\/em> at a Time<\/td>\r\n<\/tr>\r\n
Probability addition rule<\/td>\r\n$P(E\\cup F)=P(E)+P(F)-P(E\\cap F)$<\/em><\/td>\r\nThe probability of the union of two events<\/td>\r\n<\/tr>\r\n
The complement rule<\/td>\r\n$P(E^c)= 1-P(E)$<\/em><\/td>\r\n<\/td>\r\n<\/tr>\r\n
Conditional probability<\/td>\r\nP(E | F) = $\\frac{P(E\\cap F)}{P(F)}$<\/em><\/td>\r\nThe probability of E<\/em> given F<\/em><\/td>\r\n<\/tr>\r\n
Independence test<\/td>\r\n$P(E\\cap F) = P(E)\\,P(F)$<\/em><\/td>\r\nTwo Events E<\/em> and F<\/em> are independent.<\/td>\r\n<\/tr>\r\n
Binomial Probability Theorem<\/td>\r\nP(n, k; p) = nCkpk<\/sup>qn - k<\/sup><\/em><\/td>\r\np<\/em> denotes the probability of success and q<\/em> = (1 \u2212 p<\/em>) the probability of failure.<\/td>\r\n<\/tr>\r\n
Bayes' Formula<\/td>\r\n$P(A_i\\,|\\, E) = \\frac{P(A_i)\\, P(E\\,|\\, A_i)}{P(A_1)\\, P(E\\,|\\, A_1)+P(A_2)\\, P(E\\,|\\, A_2)+\\,\\cdots\\,+P(A_n)\\, P(E\\,|\\, A_n)}$<\/em><\/td>\r\nLet S be a sample space that is divided into n partitions, A1<\/sub>, A2<\/sub>, . . . An <\/sub><\/em>and E <\/em>is any event in S.<\/td>\r\n<\/tr>\r\n
Expected Value<\/td>\r\nExpected Value = x1<\/sub>p(x1<\/sub>) + x2<\/sub>p(x2<\/sub>) + x3<\/sub>p(x3<\/sub>) + \u00b7\u00b7\u00b7 + xn<\/sub>p(xn<\/sub>)<\/em><\/td>\r\n$p(x_1) \\text{ is the probability of } x_1, \\quad p(x_2) \\text{ is the probability of } x_2, \\quad \\text{and more}$<\/span><\/em><\/td>\r\n<\/tr>\r\n
Markov Chains<\/td>\r\n$\\left[\\begin{array}{cc} x_1 & x_2\\end{array}\\right] \\left[\\begin{array}{cc} y_1 & y_3 \\\\y_2 & y_4\\end{array}\\right]=\\left[\\begin{array}{cc} (x_1)(y_1) + (x_2)(y_2) & (x_1)(y_3) + (x_2)(y_4)\\end{array}\\right]$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
Equilibrium vector<\/td>\r\nET<\/em> = E<\/em>\r\n\r\n$E=\\left[\\begin{array}{cc} e & 1-e\\end{array}\\right]$<\/span><\/span><\/td>\r\nThe system is in steady-state or state of equilibrium in the long run. T<\/em> is the transition matrix and E<\/em> is the equilibrium vector.<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"
<\/div>\n\n\n\n
Term<\/strong><\/td>\nFormula<\/strong><\/td>\nExplanation<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Ratio<\/td>\na <\/em>: b<\/em> <\/span><\/td>\n<\/td>\n<\/tr>\n
Proportion<\/td>\n\"\frac{a}{b}=\frac{c}{d}\"<\/span><\/td>\nAn equation with a ratio (or rate) on two sides, in which the two ratios are equal.<\/td>\n<\/tr>\n
Cross-product rule<\/td>\n\"\frac{a}{b} = \frac{c}{d} \quad \text{equals} \quad a \times d = c \times b\"<\/span><\/td>\nMultiplying along two diagonals and solving for the unknown.<\/td>\n<\/tr>\n
Percent proportion method<\/td>\n\"\frac{\text{Part ("is")}}{\text{Whole ("of")}} = \frac{\%}{100}\"<\/span><\/td>\n<\/td>\n<\/tr>\n
Percent increase<\/td>\n= \"\frac{New\;value-Original\;value}{Original\;value}\"<\/span><\/td>\n<\/td>\n<\/tr>\n
Percent decrease<\/td>\n= \"\frac{Original\;value-New\;value}{Original\;value}\"<\/span><\/td>\n<\/td>\n<\/tr>\n
Linear equation: slope-intercept form<\/td>\ny = m<\/em>x + b<\/em><\/td>\nm\u00a0<\/em>is the slope and b <\/em>is the constant.<\/td>\n<\/tr>\n
Slope of a linear equation<\/td>\n\"m = \frac{y_2 - y_1}{x_2 - x_1}\"<\/span><\/td>\n\"(x_1, y_1) \text{ and } (x_2, y_2) \text{ are two points.}\"<\/em><\/td>\n<\/tr>\n
Linear equation: standard form<\/td>\nAx + By = C<\/span><\/td>\n\u00a0<\/em><\/td>\n<\/tr>\n
Slope of a linear equation in standard form<\/td>\n\"m = -\frac{A}{B}\"<\/span><\/td>\n\u00a0<\/em><\/td>\n<\/tr>\n
Linear equation: Point-slope formula<\/td>\n\"y-y_1=m(x-x_1)\"<\/span><\/td>\n\"(x_1, y_1) \text{ is a point.}\"<\/span><\/em><\/td>\n<\/tr>\n
Average rate of change<\/td>\n\"\frac{Change\;of\;Output}{Change\;of\;Input}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{ x_2-x_1}\"<\/span><\/td>\nThe average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.<\/td>\n<\/tr>\n
Quadratic function: standard form<\/td>\n\"f(x)=ax^2+bx+c\"<\/span><\/td>\n\u00a0<\/span><\/em><\/td>\n<\/tr>\n
Quadratic function: vertex form<\/td>\n\"f(x)=a(x-h)^2+k\"<\/span><\/td>\n\"(h, k) \text{ is the vertex point.}\"<\/span><\/em><\/td>\n<\/tr>\n
Vertex of a quadratic function<\/td>\n\n

\"h=-\frac{b}{2a}\",\u00a0 \u00a0\"k=f(h)=f(\frac{-b}{2a})\"<\/p>\n<\/td>\n

\u00a0<\/span><\/em><\/td>\n<\/tr>\n
Quadratic formula<\/td>\n\"x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}\"<\/td>\nIt gives the horizontal intercepts.<\/td>\n<\/tr>\n
Exponential growth or decay function<\/td>\n\"f(x)=a(1+r)^x\"\u00a0 \u00a0or\u00a0 \u00a0\"f(x)=ab^x\"\u00a0 \u00a0where \"b=1+r\"<\/td>\na<\/em> is the initial or starting value, r<\/em> is the percent growth or decay rate, and b<\/em> is the growth factor.<\/p>\n

\"\text{Note: if we have a decay rate, } 1 + r \text{ should be changed to } 1 - r \text{.}\"<\/span><\/td>\n<\/tr>\n

Compound Interest Formula<\/td>\n\"A(t)=a(1+\frac{r}{k})^{kt}\"<\/td>\nr<\/em> is the annual percentage rate (APR), also called the nominal rate, and k<\/em> is the number of compounding periods in one year<\/td>\n<\/tr>\n
Continuous Growth Formula<\/td>\nf(x) = \"ae^{rx}\"<\/span><\/td>\n\"\text{Note: if we have a decay rate, } r \text{ should be changed to } - r \text{.}\"<\/span><\/em><\/td>\n<\/tr>\n
Conversion to log form<\/td>\nba<\/sup><\/em> = c<\/em> is equivalent to the statement logb<\/em><\/sub>(c<\/em>) = a<\/em> .<\/td>\n\u00a0<\/span><\/em><\/td>\n<\/tr>\n
Properties of Logs: Inverse Properties<\/td>\nlogb<\/em><\/sub>(bx<\/sup><\/em>) = x<\/em><\/p>\n

b<\/em>logb<\/sub>x<\/em><\/sup> = x<\/em><\/td>\n

\u00a0<\/span><\/em><\/td>\n<\/tr>\n
Properties of Logs: Exponent Property<\/td>\n\n

logb<\/em><\/sub>(Ar<\/sup><\/em>) = r<\/em>logb<\/em><\/sub>(A<\/em>)<\/p>\n<\/td>\n

\u00a0<\/span><\/em><\/td>\n<\/tr>\n
Properties of Logs: Change of Base<\/td>\n\n

\"\log_{b}(A)=\frac{\log_{c}(A)}{\log_{c}(b)}\"<\/p>\n<\/td>\n

\u00a0<\/span><\/em><\/td>\n<\/tr>\n
Sum of Logs Property<\/td>\n\n

\"\log_{b}(A)+\log_{b}(C)=\log_b(AC)\"<\/p>\n<\/td>\n

\u00a0<\/span><\/em><\/td>\n<\/tr>\n
Difference of Logs Property<\/td>\n\n

\"\log_{b}(A)-log_{b}(C)=\log_{b}(\frac{A}{C})\"<\/p>\n<\/td>\n

\u00a0<\/span><\/em><\/td>\n<\/tr>\n
Half-Life (Based on standard exponential function)<\/td>\n\n

\"\frac{1}{2}=b^t\"<\/p>\n<\/td>\n

\u00a0<\/span><\/em><\/td>\n<\/tr>\n
Half-Life (Based on continuous change function)<\/td>\n\n

\"\frac{1}{2}=e^{rt}\"<\/p>\n<\/td>\n

\u00a0<\/span><\/em><\/td>\n<\/tr>\n
Doubling Time (Based on standard exponential function)<\/td>\n\n

2 = bt<\/sup><\/em><\/p>\n<\/td>\n

\u00a0<\/span><\/em><\/td>\n<\/tr>\n
Half-Life (Based on continuous change function)<\/td>\n\n

\"2=e^{rt}\"<\/p>\n<\/td>\n

\u00a0<\/span><\/em><\/td>\n<\/tr>\n
Subset<\/td>\n\n

\"A\subseteq B\"<\/p>\n<\/td>\n

set A is a subset of a set B if every member of A is also a member of B.<\/td>\n<\/tr>\n
Union of two sets<\/td>\n\n

\"A\cup B\"<\/p>\n<\/td>\n

The set of all elements that are either in A or in B, or in both.<\/td>\n<\/tr>\n
Intersection of two sets<\/td>\n\n

\"A\cap B\"<\/p>\n<\/td>\n

The set of all elements that are common to both sets A and B.<\/td>\n<\/tr>\n
Complement of a set<\/td>\n\n

\"A^c\"<\/p>\n<\/td>\n

The set consists of elements in the universal set U that are not in A.<\/td>\n<\/tr>\n
Permutations<\/td>\n\n

n<\/em>Pr<\/em> = \"\frac{n!}{(n-r)!}\"<\/p>\n<\/td>\n

The Number of Permutations of n<\/em> Objects Taken r<\/em> at a Time.<\/td>\n<\/tr>\n
Circular Permutations<\/td>\n\n

(n \u2212 1)!<\/em><\/p>\n<\/td>\n

The number of permutations of n<\/em> elements in a circle.<\/td>\n<\/tr>\n
Permutations with Similar Elements<\/td>\n\n

\"\frac{n!}{r_1!r_2!...r_k!}\"<\/em><\/p>\n<\/td>\n

The number of permutations of n<\/em> elements taken n<\/em> at a time, with r<\/em>1<\/sub> elements of one kind, r<\/em>2<\/sub> elements of another kind, and so on.<\/td>\n<\/tr>\n
Combinations<\/td>\nn<\/em>Cr<\/em> = \"\frac{n!}{(n-r)!r!}\"<\/td>\nThe Number of Combinations of n<\/em> Objects Taken r<\/em> at a Time<\/td>\n<\/tr>\n
Probability addition rule<\/td>\n\"P(E\cup F)=P(E)+P(F)-P(E\cap F)\"<\/em><\/td>\nThe probability of the union of two events<\/td>\n<\/tr>\n
The complement rule<\/td>\n\"P(E^c)= 1-P(E)\"<\/em><\/td>\n<\/td>\n<\/tr>\n
Conditional probability<\/td>\nP(E | F) = \"\frac{P(E\cap F)}{P(F)}\"<\/em><\/td>\nThe probability of E<\/em> given F<\/em><\/td>\n<\/tr>\n
Independence test<\/td>\n\"P(E\cap F) = P(E)\,P(F)\"<\/em><\/td>\nTwo Events E<\/em> and F<\/em> are independent.<\/td>\n<\/tr>\n
Binomial Probability Theorem<\/td>\nP(n, k; p) = nCkpk<\/sup>qn – k<\/sup><\/em><\/td>\np<\/em> denotes the probability of success and q<\/em> = (1 \u2212 p<\/em>) the probability of failure.<\/td>\n<\/tr>\n
Bayes’ Formula<\/td>\n\"P(A_i\,|\, E) = \frac{P(A_i)\, P(E\,|\, A_i)}{P(A_1)\, P(E\,|\, A_1)+P(A_2)\, P(E\,|\, A_2)+\,\cdots\,+P(A_n)\, P(E\,|\, A_n)}\"<\/em><\/td>\nLet S be a sample space that is divided into n partitions, A1<\/sub>, A2<\/sub>, . . . An <\/sub><\/em>and E <\/em>is any event in S.<\/td>\n<\/tr>\n
Expected Value<\/td>\nExpected Value = x1<\/sub>p(x1<\/sub>) + x2<\/sub>p(x2<\/sub>) + x3<\/sub>p(x3<\/sub>) + \u00b7\u00b7\u00b7 + xn<\/sub>p(xn<\/sub>)<\/em><\/td>\n\"p(x_1) \text{ is the probability of } x_1, \quad p(x_2) \text{ is the probability of } x_2, \quad \text{and more}\"<\/span><\/em><\/td>\n<\/tr>\n
Markov Chains<\/td>\n\"\left[\begin{array}{cc} x_1 & x_2\end{array}\right] \left[\begin{array}{cc} y_1 & y_3 \\y_2 & y_4\end{array}\right]=\left[\begin{array}{cc} (x_1)(y_1) + (x_2)(y_2) & (x_1)(y_3) + (x_2)(y_4)\end{array}\right]\"<\/td>\n<\/td>\n<\/tr>\n
Equilibrium vector<\/td>\nET<\/em> = E<\/em><\/p>\n

\"E=\left[\begin{array}{cc} e & 1-e\end{array}\right]\"<\/span><\/span><\/td>\n

The system is in steady-state or state of equilibrium in the long run. T<\/em> is the transition matrix and E<\/em> is the equilibrium vector.<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":2,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["haghighi"],"pb_section_license":""},"front-matter-type":[],"contributor":[62],"license":[],"class_list":["post-19","front-matter","type-front-matter","status-publish","hentry","contributor-haghighi"],"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/wp-json\/pressbooks\/v2\/front-matter\/19","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/wp-json\/pressbooks\/v2\/front-matter"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/wp-json\/wp\/v2\/types\/front-matter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/wp-json\/wp\/v2\/users\/2"}],"version-history":[{"count":30,"href":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/wp-json\/pressbooks\/v2\/front-matter\/19\/revisions"}],"predecessor-version":[{"id":224,"href":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/wp-json\/pressbooks\/v2\/front-matter\/19\/revisions\/224"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/wp-json\/pressbooks\/v2\/front-matter\/19\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/wp-json\/wp\/v2\/media?parent=19"}],"wp:term":[{"taxonomy":"front-matter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/wp-json\/pressbooks\/v2\/front-matter-type?post=19"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/wp-json\/wp\/v2\/contributor?post=19"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/wp-json\/wp\/v2\/license?post=19"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}