# F | Mathematical Phrases, Symbols, and Formulas

### F | Mathematical Phrases, Symbols, and Formulas

#### 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(n-1)(n-2)...\left(1\right)\text{}$$0!=1\text{}$

#### Formula 2: Combinations

$\left(\begin{array}{l}n\\ r\end{array}\right)=\frac{n!}{(n-r)!r!}$#### Formula 3: Binomial Distribution

$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}B(n,p)$$P(X=x)=\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(p)$$P(X=x)={q}^{x-1}p$, for $x=1,2,3,...$

#### Formula 5: Hypergeometric Distribution

$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}H(r,b,n)$$P\text{(}X=x\text{)}=\left(\frac{\left(\genfrac{}{}{0ex}{}{r}{x}\right)\left(\genfrac{}{}{0ex}{}{b}{n-x}\right)}{\left(\genfrac{}{}{0ex}{}{r+b}{n}\right)}\right)$

#### Formula 6: Poisson Distribution

$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}P(\mu )$$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(a,b)$$f(X)=\frac{1}{b-a}$, $a<x<b$

#### Formula 8: Exponential Distribution

$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}Exp(m)$$f(x)=m{e}^{-mx}m>0,x\ge 0$

#### Formula 9: Normal Distribution

$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}N(\mu ,{\sigma}^{2})$$f\text{(}x\text{)}=\frac{1}{\sigma \sqrt{2\pi}}{e}^{\frac{{-(x-\mu )}^{2}}{{2\sigma}^{2}}}$ , $\phantom{\rule{12pt}{0ex}}\u2013\infty <x<\infty $

#### Formula 10: Gamma Function

$\Gamma (z)=\underset{\infty}{\overset{0}{{{\displaystyle \int}}^{\text{}}}}{x}^{z-1}{e}^{-x}dx$$z>0$$\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi}$

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

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

#### 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{-(n+1)}{2}}\Gamma \left(\frac{n+1}{2}\right)}{\sqrt{\mathrm{n\pi}}\Gamma \left(\frac{n}{2}\right)}$

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

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

#### Formula 12: Chi-Square Distribution

$X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}{{\rm X}}_{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(n),df(d)}$$df(n)\phantom{\rule{2px}{0ex}}=\phantom{\rule{2px}{0ex}}$degrees of freedom for the numerator

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

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

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

#### Symbols and Their Meanings

Chapter (1st used) | Symbol | Spoken | Meaning |
---|---|---|---|

Sampling and Data | $\sqrt{\begin{array}{c}\text{}\\ \text{}\end{array}}$ | The square root of | same |

Sampling and Data | $\pi $ | Pi | 3.14159… (a specific number) |

Descriptive Statistics | Q_{1} |
Quartile one | the first quartile |

Descriptive Statistics | Q_{2} |
Quartile two | the second quartile |

Descriptive Statistics | Q_{3} |
Quartile three | the third quartile |

Descriptive Statistics | IQR |
interquartile range | Q_{3} – Q_{1} = IQR |

Descriptive Statistics | $\overline{x}$ | x-bar | sample mean |

Descriptive Statistics | $\mu $ | mu | population mean |

Descriptive Statistics | s s _{x}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 | $P(\mathit{\text{A}}\text{|}\mathit{\text{B}})$ | probability of A given B | prob. of A occurring given B has occurred |

Probability Topics | $P(A\text{OR}B)$ | prob. of A or B | prob. of A or B or both occurring |

Probability Topics | $P(A\text{AND}B)$ | 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 | G_{1} |
green on first pick | same |

Probability Topics | P(G_{1}) |
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 $; $\hat{p}$ | p-prime; p-hat |
sample proportion of success |

Confidence Intervals | $q\prime $; $\hat{q}$ | 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 | ${{\rm X}}^{2}$ | 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 | $\hat{y}$ | 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 |