Tài liệu

# Appendix F: Mathematical Phrases, Symbols, and Formulas

Mathematics and Statistics

# English Phrases Written Mathematically

 When the English says: Interpret this as: X is at least 4. X ≥ 4 The minimum of X is 4. X ≥ 4 X is no less than 4. X ≥ 4 X is greater than or equal to 4. X ≥ 4 X is at most 4. X ≤ 4 The maximum of X is 4. X ≤ 4 X is no more than 4. X ≤ 4 X is less than or equal to 4. X ≤ 4 X does not exceed 4. X ≤ 4 X is greater than 4. X > 4 X is more than 4. X > 4 X exceeds 4. X > 4 X is less than 4. X < 4 There are fewer X than 4. X < 4 X is 4. X = 4 X is equal to 4. X = 4 X is the same as 4. X = 4 X is not 4. X ≠ 4 X is not equal to 4. X ≠ 4 X is not the same as 4. X ≠ 4 X is different than 4. X ≠ 4

# Formulas

Formula 1: Factorial $n!=n\left(n-1\right)\left(n-2\right)...\left(1\right)\text{}$

$0!=1\text{}$

Formula 2: Combinations$\left(\begin{array}{l}n\\ r\end{array}\right)=\frac{n!}{\left(n-r\right)!r!}$

Formula 3: Binomial Distribution$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}B\left(n,p\right)$

$P\left(X=x\right)=\left(\begin{array}{c}n\\ x\end{array}\right){p}^{x}{q}^{n-x}$, for $x=0,1,2,...,n$

Formula 4: Geometric Distribution$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}G\left(p\right)$

$P\left(X=x\right)={q}^{x-1}p$, for $x=1,2,3,...$

Formula 5: Hypergeometric Distribution$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}H\left(r,b,n\right)$

$P\text{(}X=x\text{)}=\left(\frac{\left(\genfrac{}{}{0}{}{r}{x}\right)\left(\genfrac{}{}{0}{}{b}{n-x}\right)}{\left(\genfrac{}{}{0}{}{r+b}{n}\right)}\right)$

Formula 6: Poisson Distribution$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}P\left(\mu \right)$

$P\text{(}X=x\text{)}=\frac{{\mu }^{x}{e}^{-\mu }}{x!}$

Formula 7: Uniform Distribution$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}U\left(a,b\right)$

$f\left(X\right)=\frac{1}{b-a}$, $a

Formula 8: Exponential Distribution$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}Exp\left(m\right)$

$f\left(x\right)=m{e}^{-mx}m>0,x\ge 0$

Formula 9: Normal Distribution$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}N\left(\mu ,{\sigma }^{2}\right)$

$f\text{(}x\text{)}=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{{-\left(x-\mu \right)}^{2}}{{2\sigma }^{2}}}$ , $\phantom{\rule{12pt}{0ex}}–\infty

Formula 10: Gamma Function$\Gamma \left(z\right)=\underset{\infty }{\overset{0}{{\int }^{\text{​}}}}{x}^{z-1}{e}^{-x}dx$ $z>0$

$\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }$

$\Gamma \left(m+1\right)=m!$ for $m$, a nonnegative integer

otherwise: $\Gamma \left(a+1\right)=a\Gamma \left(a\right)$

Formula 11: Student's t-distribution$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}{t}_{df}$

$f\text{(}x\text{)}=\frac{{\left(1+\frac{{x}^{2}}{n}\right)}^{\frac{-\left(n+1\right)}{2}}\Gamma \left(\frac{n+1}{2}\right)}{\sqrt{\mathrm{n\pi }}\Gamma \left(\frac{n}{2}\right)}$

$X=\frac{Z}{\sqrt{\frac{Y}{n}}}$

$Z\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}N\left(0,1\right),\phantom{\rule{2px}{0ex}}Y\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}{Χ}_{df}^{2}$, $n$ = degrees of freedom

Formula 12: Chi-Square Distribution$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}{Χ}_{df}^{2}$

$f\text{(}x\text{)}=\frac{{x}^{\frac{n-2}{2}}{e}^{\frac{-x}{2}}}{{2}^{\frac{n}{2}}\Gamma \left(\frac{n}{2}\right)}$, $x>0$ , $n$ = positive integer and degrees of freedom

Formula 13: F Distribution$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}{F}_{df\left(n\right),df\left(d\right)}$

$df\left(n\right)\phantom{\rule{2px}{0ex}}=\phantom{\rule{2px}{0ex}}$degrees of freedom for the numerator

$df\left(d\right)\phantom{\rule{2px}{0ex}}=\phantom{\rule{2px}{0ex}}$degrees of freedom for the denominator

$f\left(x\right)=\frac{\Gamma \left(\frac{u+v}{2}\right)}{\Gamma \left(\frac{u}{2}\right)\Gamma \left(\frac{v}{2}\right)}{\left(\frac{u}{v}\right)}^{\frac{u}{2}}{x}^{\left(\frac{u}{2}-1\right)}\left[1+\left(\frac{u}{v}\right){x}^{-0.5\left(u+v\right)}\right]$

$X=\frac{{Y}_{u}}{{W}_{v}}$, $Y$, $W$ are chi-square

# Symbols and Their Meanings

 Chapter (1st used) Symbol Spoken Meaning Sampling and Data The square root of same Sampling and Data $\pi$ Pi 3.14159… (a specific number) Descriptive Statistics Q1 Quartile one the first quartile Descriptive Statistics Q2 Quartile two the second quartile Descriptive Statistics Q3 Quartile three the third quartile Descriptive Statistics IQR interquartile range Q3 – Q1 = IQR Descriptive Statistics $\overline{x}$ x-bar sample mean Descriptive Statistics $\mu$ mu population mean Descriptive Statistics s sx sx s sample standard deviation Descriptive Statistics ${s}^{2}$ ${s}_{x}^{2}$ s squared sample variance Descriptive Statistics $\sigma$ ${\sigma }_{x}$ σx sigma population standard deviation Descriptive Statistics ${\sigma }^{2}$ ${\sigma }_{x}^{2}$ sigma squared population variance Descriptive Statistics $\Sigma$ capital sigma sum Probability Topics $\left\{\right\}$ brackets set notation Probability Topics $S$ S sample space Probability Topics $A$ Event A event A Probability Topics $P\left(A\right)$ probability of A probability of A occurring Probability Topics $P\left(\mathit{\text{A}}\text{|}\mathit{\text{B}}\right)$ probability of A given B prob. of A occurring given B has occurred Probability Topics prob. of A or B prob. of A or B or both occurring Probability Topics prob. of A and B prob. of both A and B occurring (same time) Probability Topics A′ A-prime, complement of A complement of A, not A Probability Topics P(A') prob. of complement of A same Probability Topics G1 green on first pick same Probability Topics P(G1) prob. of green on first pick same Discrete Random Variables PDF prob. distribution function same Discrete Random Variables X X the random variable X Discrete Random Variables X ~ the distribution of X same Discrete Random Variables B binomial distribution same Discrete Random Variables G geometric distribution same Discrete Random Variables H hypergeometric dist. same Discrete Random Variables P Poisson dist. same Discrete Random Variables $\lambda$ Lambda average of Poisson distribution Discrete Random Variables $\ge$ greater than or equal to same Discrete Random Variables $\le$ less than or equal to same Discrete Random Variables = equal to same Discrete Random Variables ≠ not equal to same Continuous Random Variables f(x) f of x function of x Continuous Random Variables pdf prob. density function same Continuous Random Variables U uniform distribution same Continuous Random Variables Exp exponential distribution same Continuous Random Variables k k critical value Continuous Random Variables f(x) = f of x equals same Continuous Random Variables m m decay rate (for exp. dist.) The Normal Distribution N normal distribution same The Normal Distribution z z-score same The Normal Distribution Z standard normal dist. same The Central Limit Theorem CLT Central Limit Theorem same The Central Limit Theorem $\overline{X}$ X-bar the random variable X-bar The Central Limit Theorem ${\mu }_{x}$ mean of X the average of X The Central Limit Theorem ${\mu }_{\overline{x}}$ mean of X-bar the average of X-bar The Central Limit Theorem ${\sigma }_{x}$ standard deviation of X same The Central Limit Theorem ${\sigma }_{\overline{x}}$ standard deviation of X-bar same The Central Limit Theorem $\Sigma X$ sum of X same The Central Limit Theorem $\Sigma x$ sum of x same Confidence Intervals CL confidence level same Confidence Intervals CI confidence interval same Confidence Intervals EBM error bound for a mean same Confidence Intervals EBP error bound for a proportion same Confidence Intervals t Student's t-distribution same Confidence Intervals df degrees of freedom same Confidence Intervals ${t}_{\frac{\alpha }{2}}$ student t with a/2 area in right tail same Confidence Intervals $p\prime$; $\stackrel{^}{p}$ p-prime; p-hat sample proportion of success Confidence Intervals $q\prime$; $\stackrel{^}{q}$ q-prime; q-hat sample proportion of failure Hypothesis Testing ${H}_{0}$ H-naught, H-sub 0 null hypothesis Hypothesis Testing ${H}_{a}$ H-a, H-sub a alternate hypothesis Hypothesis Testing ${H}_{1}$ H-1, H-sub 1 alternate hypothesis Hypothesis Testing $\alpha$ alpha probability of Type I error Hypothesis Testing $\beta$ beta probability of Type II error Hypothesis Testing $\overline{X1}-\overline{X2}$ X1-bar minus X2-bar difference in sample means Hypothesis Testing ${\mu }_{1}-{\mu }_{2}$ mu-1 minus mu-2 difference in population means Hypothesis Testing ${{P}^{\prime }}_{1}-{{P}^{\prime }}_{2}$ P1-prime minus P2-prime difference in sample proportions Hypothesis Testing ${p}_{1}-{p}_{2}$ p1 minus p2 difference in population proportions Chi-Square Distribution ${Χ}^{2}$ Ky-square Chi-square Chi-Square Distribution $O$ Observed Observed frequency Chi-Square Distribution $E$ Expected Expected frequency Linear Regression and Correlation y = a + bx y equals a plus b-x equation of a line Linear Regression and Correlation $\stackrel{^}{y}$ y-hat estimated value of y Linear Regression and Correlation $r$ correlation coefficient same Linear Regression and Correlation $\epsilon$ error same Linear Regression and Correlation SSE Sum of Squared Errors same Linear Regression and Correlation 1.9s 1.9 times s cut-off value for outliers F-Distribution and ANOVA F F-ratio F-ratio
Tải về
Đánh giá:
0 dựa trên 0 đánh giá

#### Tuyển tập sử dụng module này

Nội dung cùng tác giả

Nội dung tương tự