# 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
Table F1

#### Formula 1: Factorial

$n!=n(n−1)(n−2)...(1)n!=n(n−1)(n−2)...(1)$

$0!=10!=1$

#### Formula 2: Combinations

$(nr)=n!(n−r)!r!(nr)=n!(n−r)!r!$

#### Formula 3: Binomial Distribution

$X~B(n,p)X~B(n,p)$

$P(X=x)=(nx)pxqn−xP(X=x)=(nx)pxqn−x$, for $x=0,1,2,...,nx=0,1,2,...,n$

#### Formula 4: Geometric Distribution

$X~G(p)X~G(p)$

$P(X=x)=qx−1pP(X=x)=qx−1p$, for $x=1,2,3,...x=1,2,3,...$

#### Formula 5: Hypergeometric Distribution

$X~H(r,b,n)X~H(r,b,n)$

$P(X=x)=((rx)(bn−x)(r+bn))P(X=x)=((rx)(bn−x)(r+bn))$

#### Formula 6: Poisson Distribution

$X~P(μ)X~P(μ)$

$P(X=x)=μxe−μx!P(X=x)=μxe−μx!$

#### Formula 7: Uniform Distribution

$X~U(a,b)X~U(a,b)$

$f(X)=1b−af(X)=1b−a$, $a

#### Formula 8: Exponential Distribution

$X~Exp(m)X~Exp(m)$

$f(x)=me−mxm>0,x≥0f(x)=me−mxm>0,x≥0$

#### Formula 9: Normal Distribution

$X~N(μ,σ2)X~N(μ,σ2)$

$f(x)=1σ2πe−(x−μ)22σ2f(x)=1σ2πe−(x−μ)22σ2$ , $–∞

#### Formula 10: Gamma Function

$Γ(z)=∫​∞0xz−1e−xdxΓ(z)=∫​∞0xz−1e−xdx$$z>0z>0$

$Γ(12)=πΓ(12)=π$

$Γ(m+1)=m!Γ(m+1)=m!$ for $mm$, a nonnegative integer

otherwise: $Γ(a+1)=aΓ(a)Γ(a+1)=aΓ(a)$

#### Formula 11: Student's t-distribution

$X~tdfX~tdf$

$f(x)=(1+x2n)−(n+1)2Γ(n+12)nπΓ(n2)f(x)=(1+x2n)−(n+1)2Γ(n+12)nπΓ(n2)$

$X=ZYnX=ZYn$

$Z~N(0,1),Y~Χdf2Z~N(0,1),Y~Χdf2$, $nn$ = degrees of freedom

#### Formula 12: Chi-Square Distribution

$X~Χdf2X~Χdf2$

$f(x)=xn−22e−x22n2Γ(n2)f(x)=xn−22e−x22n2Γ(n2)$, $x>0x>0$ , $nn$ = positive integer and degrees of freedom

#### Formula 13: F Distribution

$X~Fdf(n),df(d)X~Fdf(n),df(d)$

$df(n)=df(n)=$degrees of freedom for the numerator

$df(d)=df(d)=$degrees of freedom for the denominator

$f(x)=Γ(u+v2)Γ(u2)Γ(v2)(uv)u2x(u2−1)[1+(uv)x−0.5(u+v)]f(x)=Γ(u+v2)Γ(u2)Γ(v2)(uv)u2x(u2−1)[1+(uv)x−0.5(u+v)]$

$X=YuWvX=YuWv$, $YY$, $WW$ are chi-square

#### Symbols and Their Meanings

Chapter (1st used) Symbol Spoken Meaning
Sampling and Data The square root of same
Sampling and Data $ππ$ 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 Q3Q1 = IQR
Descriptive Statistics $x¯x¯$ x-bar sample mean
Descriptive Statistics $μμ$ mu population mean
Descriptive Statistics s sx sx s sample standard deviation
Descriptive Statistics $s2s2$$sx2sx2$ s squared sample variance
Descriptive Statistics $σσ$$σxσx$σx sigma population standard deviation
Descriptive Statistics $σ2σ2$$σx2σx2$ sigma squared population variance
Descriptive Statistics $ΣΣ$ capital sigma sum
Probability Topics ${}{}$ brackets set notation
Probability Topics $SS$ S sample space
Probability Topics $AA$ Event A event A
Probability Topics $P(A)P(A)$ probability of A probability of A occurring
Probability Topics $P(A|B)P(A|B)$ 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 $λλ$ Lambda average of Poisson distribution
Discrete Random Variables $≥≥$ greater than or equal to same
Discrete Random Variables $≤≤$ 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 $X¯X¯$ X-bar the random variable X-bar
The Central Limit Theorem $μxμx$ mean of X the average of X
The Central Limit Theorem $μx¯μx¯$ mean of X-bar the average of X-bar
The Central Limit Theorem $σxσx$ standard deviation of X same
The Central Limit Theorem $σx¯σx¯$ standard deviation of X-bar same
The Central Limit Theorem $ΣXΣX$ sum of X same
The Central Limit Theorem $ΣxΣx$ 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 $tα2tα2$ student t with a/2 area in right tail same
Confidence Intervals $p′p′$; $p^p^$ p-prime; p-hat sample proportion of success
Confidence Intervals $q′q′$; $q^q^$ q-prime; q-hat sample proportion of failure
Hypothesis Testing $H0H0$ H-naught, H-sub 0 null hypothesis
Hypothesis Testing $HaHa$ H-a, H-sub a alternate hypothesis
Hypothesis Testing $H1H1$ H-1, H-sub 1 alternate hypothesis
Hypothesis Testing $αα$ alpha probability of Type I error
Hypothesis Testing $ββ$ beta probability of Type II error
Hypothesis Testing $X1¯−X2¯X1¯−X2¯$ X1-bar minus X2-bar difference in sample means
Hypothesis Testing $μ1−μ2μ1−μ2$ mu-1 minus mu-2 difference in population means
Hypothesis Testing $P′1−P′2P′1−P′2$ P1-prime minus P2-prime difference in sample proportions
Hypothesis Testing $p1−p2p1−p2$ p1 minus p2 difference in population proportions
Chi-Square Distribution $Χ2Χ2$ Ky-square Chi-square
Chi-Square Distribution $OO$ Observed Observed frequency
Chi-Square Distribution $EE$ Expected Expected frequency
Linear Regression and Correlation y = a + bx y equals a plus b-x equation of a line
Linear Regression and Correlation $y^y^$ y-hat estimated value of y
Linear Regression and Correlation $rr$ correlation coefficient same
Linear Regression and Correlation $εε$ 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
Table F2