World Of Mathematics: August 2013

Thursday, August 22, 2013

Probability and statistics symbols

Statistical Symbols

Probability and statistics symbols table and definitions.

Probability and statistics symbols table

Symbol	Symbol Name	Meaning / definition	Example
P(A)	probability function	probability of event A	P(A) = 0.5
P(A ∩ B)	probability of events intersection	probability that of events A and B	P(A∩B) = 0.5
P(A ∪ B)	probability of events union	probability that of events A or B	P(A∪B) = 0.5
P(A \| B)	conditional probability function	probability of event A given event B occured	P(A \| B) = 0.3
f (x)	probability density function (pdf)	P(a ≤ x ≤ b) = ∫ f (x) dx
F(x)	cumulative distribution function (cdf)	F(x) = P(X ≤ x)
μ	population mean	mean of population values	μ = 10
E(X)	expectation value	expected value of random variable X	E(X) = 10
E(X \| Y)	conditional expectation	expected value of random variable X given Y	E(X \| Y=2) = 5
var(X)	variance	variance of random variable X	var(X) = 4
σ²	variance	variance of population values	σ² = 4
std(X)	standard deviation	standard deviation of random variable X	std(X) = 2
σ_X	standard deviation	standard deviation value of random variable X	σ_X = 2
	median	middle value of random variable x
cov(X,Y)	covariance	covariance of random variables X and Y	cov(X,Y) = 4
corr(X,Y)	correlation	correlation of random variables X and Y	corr(X,Y) = 0.6
ρ_X,Y	correlation	correlation of random variables X and Y	ρ_X,Y = 0.6
∑	summation	summation - sum of all values in range of series
∑∑	double summation	double summation
Mo	mode	value that occurs most frequently in population
MR	mid-range	MR = (x_max+x_min)/2
Md	sample median	half the population is below this value
Q₁	lower / first quartile	25% of population are below this value
Q₂	median / second quartile	50% of population are below this value = median of samples
Q₃	upper / third quartile	75% of population are below this value
x	sample mean	average / arithmetic mean	x = (2+5+9) / 3 = 5.333
s ²	sample variance	population samples variance estimator	s² = 4
s	sample standard deviation	population samples standard deviation estimator	s = 2
z_x	standard score	z_x = (x-x) / s_x

X ~
distribution of X
distribution of random variable X
X ~ N(0,3)
N(μ,σ²)
normal distribution
gaussian distribution
X ~ N(0,3)
U(a,b)
uniform distribution
equal probability in range a,b
X ~ U(0,3)
exp(λ)
exponential distribution
f (x) = λe^-λx , x≥0

gamma(c, λ)
gamma distribution

f (x) = λ c x^c-1e^-λx / Γ(c), x≥0

χ²(k)
chi-square distribution

f (x) = x^k^/2-1e^-x/2 / ( 2^k/2Γ(k/2) )

F (k₁, k₂)
F distribution

Bin(n,p)
binomial distribution

f (k) = _nC_k p^k(1-p)^n-k

Poisson(λ)
Poisson distribution

f (k) = λ^ke^-λ / k!

Geom(p)
geometric distribution

f (k) = p(1-p)^k

HG(N,K,n)
hyper-geometric distribution

Bern(p)
Bernoulli distribution

Combinatorics Symbols

Symbol	Symbol Name	Meaning / definition	Example
n!	factorial	n! = 1·2·3·...·n	5! = 1·2·3·4·5 = 120
_nP_k	permutation	$_{n}P_{k}=\frac{n!}{(n-k)!}$	₅P₃ = 5! / (5-3)! = 60
_nC_k	combination	$_{n}C_{k}=\binom{n}{k}=\frac{n!}{k!(n-k)!}$	₅C₃ = 5!/[3!(5-3)!]=10

Statistical Symbols

Statistical Symbols

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Alphabetical Statistical Symbols

Symbol	Text Equivalent	Meaning	Formula	Link to Glossary (if appropriate)
a		Y- intercept of least square regression line	, for line	Regression: y on x
b		Slope of least squares regression line	for line	Regression: y on x
	Binomial distribution with parameters n and p	Discrete probability distribution for the probability of number of successes in n independent random trials under the identical conditions.	If X follows then, , Where, , r = 0,1,2, …….,n,	Binomial Distribution
c		Confidence level		Confidence interval
ⁿC_r	n-c-r	Combinations (number of combinations of n objects taken r at a time)	, where
	n-c-r	Combinations (number of combinations of n objects taken r at a time)	, where
	Covariance between X and Y	Covariance between X & Y
CV		Coefficient of variation
df		Degree(s) of freedom
E		Maximal error tolerance	for large samples
E (f(x))	Expected value of f(x)
f		Frequency	number of times score.
F		F-distribution variable	where n₁ and n₂ are the corresponding degrees of freedom.	F-distribution, Hypothesis testing for equality of 2 variances.
or		Distribution function	.
f(x) or		Probability mass function	Depends on the distribution.
H₀	H-naught	Null hypothesis	The null hypothesis is the hypothesis about the population parameter.	Testing of hypothesis
H₁	H-one	Alternate hypothesis	An alternate hypothesis is constructed in such a way that it is the one to be accepted when the null hypothesis must be rejected.	Testing of hypothesis
IQR		Interquartile range		Measures of central tendency.
MS	M-S	Mean square		Analysis of variance (ANOVA)
n		Sample size.	number of units in a sample.
N		Population size	Number of units in the population.
	n-p-r	Permutation (number of ways to arrange in order n distinct objects taking them r at a time)	where
	n-p-r	Permutation (number of ways to arrange in order n distinct objects taking them r at a time)	, where
	p-hat	Sample proportion		Binomial distribution
	Probability of A given B	Conditional probability
	Probability of x	Probability of x
p-value		The attained level of significance.	P value is the smallest level of significance for which the observed sample statistic tells us to reject the null hypothesis.
Q		Probability of not happening of the event
Q₁	Q-one	First quartile	Median of the lower half of the data that is data below median.	Measures of central tendency
Q₂	Q-two	Second quartile Or Median	Central value of an ordered data.	Measures of central tendency
Q₃	Q-three	Third quartile	Median of the upper half of the data that is data above the median.	Measures of central tendency
R		Sample Correlation coefficient
r²	r-square	Coefficient of determination
R²	r-square	Multiple correlation coefficient
S		Sample standard deviation	for ungrouped data. for grouped data.	Measures of dispersion
S²	S-square	Sample variance	for ungrouped data. for grouped data.	Measures of dispersion
	s-e- square	Error variance
SD		Sample Standard deviation	for ungrouped data. for grouped data.
		Bowley’s coefficient of skewness		Measures of skew ness
		Pearson’s coefficient of skewness		Measures of skew ness
		Sum of squares	for ungrouped data for grouped data
t		Student’s t variable		t-distribution
t_c	t critical	The critical value for a confidence level c.	Number such that the area under the t distribution for a given number of degrees of freedom falling between and is equal to c.	Testing of hypothesis
Var(X)	Variance of X	Variance of X
X		Independent variable or explanatory variable in regression analysis	Eg. In the study of, yield obtained & the irrigation level, independent variable is, Irrigation level.
	x-bar	Arithmetic mean or Average of X scores.	for ungrouped data. for grouped data	Measures of central tendency
y		Dependent variable or response variable in regression analysis	Eg. In the study of, yield obtained & the irrigation level, dependent variable is, Yield obtained.
Z	Z-score	Standard normal variable (Normal variable with mean = 0 & SD = 1)	, where X follows Normal .	Standard normal distribution
	z critical	The critical value for a confidence level c.	Number such that the area under the standard normal curve falling between and is equal to c.	Testing of hypothesis Confidence interval