# 0.6 Mathematical phrases, symbols, and formulas

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## 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

## 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

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 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 $P\left(\mathit{\text{A}}\text{|}\mathit{\text{B}}\right)$ 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 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 $p\prime$ ; $\stackrel{^}{p}$ 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 $\overline{X1}-\overline{X2}$ X 1-bar minus X 2-bar difference in sample means
Hypothesis Testing ${\mu }_{1}-{\mu }_{2}$ mu -1 minus mu -2 difference in population means
Hypothesis Testing ${{P}^{\prime }}_{1}-{{P}^{\prime }}_{2}$ P 1-prime minus P 2-prime difference in sample proportions
Hypothesis Testing ${p}_{1}-{p}_{2}$ p 1 minus p 2 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 1.9 s 1.9 times s cut-off value for outliers
F -Distribution and ANOVA F F -ratio F -ratio

what is ch-square test
Uttam
How can I calculate the Class Mark, Relative frequency and the cumulative frequency on a frequency table?
what is the important in business planning and economics
explain the limitation and scope of statistics
mahelt
statistics is limited to use where data can be measured quantitatively. statistics scope is wider such as in economic planning, medical science etc.
Gurpreet
can you send me mcq type questions
Yas
Umar
which books are best to learn applied statistics for data science/ML
Gurpreet
A population consists of five numbers 2,3,6,8,11.consists all possible samples of size two which can be drawn with replacement from this population. calculate the S.E of sample means
A particular train reaches the destination in time in 75 per cent of the times.A person travels 5 times in that train.Find probability that he will reach the destination in time, for all the 5 times.
0.237
Amresh
umesh
p(x=5)= 5C0 p^5 q^0 solve this
Amresh
umesh
ok
umesh
5C0=1 p^5= (3/4)^5 q^0=(1/4)^0
Amresh
Hint(0.75 in time and 0.25 not in time)
kamugi
what is standard deviation?
It is the measure of the variation of certain values from the Mean (Center) of a frequency distribution of sample values for a particular Variable.
Dominic
what is the number of x
10
Elicia
Javed Arif
Jawed
how will you know if a group of data set is a sample or population
population is the whole set and the sample is the subset of population.
umair
if the data set is drawn out of a larger set it is a sample and if it is itself the whole complete set it can be treated as population.
Bhavika
hello everyone if I have the data set which contains measurements of each part during 10 years, may I say that it's the population or it's still a sample because it doesn't contain my measurements in the future? thanks
Alexander
Pls I hv a problem on t test is there anyone who can help?
Peggy
Dominic
Bhavika is right
Dominic
what is the problem peggy?
Bhavika
hi
Sandeep
Hello
hi
Bhavika
hii Bhavika
Dar
Hi eny population has a special definition. if that data set had all of characteristics of definition, that is population. otherwise that is a sample
Hoshyar
three coins are tossed. find the probability of no head
three coins are tossed consecutively or what ?
umair
umair
or .125 is the probability of getting no head when 3 coins are tossed
umair
🤣🤣🤣
Simone
what is two tailed test
if the diameter will be greater than 3 cm then the bullet will not fit in the barrel of the gun so you are bothered for both the sides.
umair
in this test you are worried on both the ends
umair
lets say you are designing a bullet for thw gun od diameter equals 3cm.if the diameter of the bullet is less than 3 cm then you wont be able to shoot it
umair
In order to apply weddles rule for numerical integration what is minimum number of ordinates
excuse me?
Gabriel
why?
didn't understand the question though.
Gabriel
which question? ?
We have rules of numerical integration like Trapezoidal rule, Simpson's 1/3 and 3/8 rules, Boole's rule and Weddle rule for n =1,2,3,4 and 6 but for n=5?
John
Someone should help me please, how can I calculate the Class Mark, Relative frequency and the cumulative frequency on a frequency table?
IJOGI
geometric mean of two numbers 4 and 16 is:
10
umair
really
iphone
quartile deviation of 8 8 8 is:
iphone
sorry 8 is the geometric mean of 4,16
umair
quartile deviation of 8 8 8 is
iphone
can you please expalin the whole question ?
umair
mcq
iphone
h
iphone
can you please post the picture of that ?
umair
how
iphone
hello
John
10 now
John
how to find out the value
can you be more specific ?
umair
yes
KrishnaReddy