Title: Theory / Important / Math / Probability / Statistics / Bulmer/
Tags: fromMath mathToDef mathToSee fromImplication closed examplesSwitchExample1 hasSeenConjDef récupéré

Name: Statistical probability
Definition: The relative frequency with which an event occurs in the long run.
Page: 1

Name: Inductive probability
Definition: The degree of belief which it is reasonable to place in a proposition on given evidence.
Page: 1
Extra: bayesian?

Definition: The limite of the ratio of times \(A\) happen.
Page: 4
Chapter: 1
Variables: \(A\) an event which may or may not happen
Notation: \(P(A)\)
Notation2: \(Prob(A)\)

Smaller0: \(P(A\text{ and } B)\)
Greater0: \(P(A)P(B\mid A)\)
Page: 16
Chapter: 2
Variables: \(A,B\) two events.
Smaller to greater: =
Name: The Law of Multiplication

Name: Statistically independent events
Definition: \(P(A\mid B)=P(A)\)
Page: 16, 17
Chapter: 2
Definition2: \(P(A,B)=P(A)P(B)\)
Conjdef: Two events \(A,B\) such that
Construction: \(P(A,B)=P(B)P(A\mid B)=P(B)P(a)\)

Name: A continuous random variable
Definition: A random variable which takes values within a certain range.
Page: 29
Chapter: 3

Name: A discrete random variable
Definition: A random variable which is not continuous.
Page: 29
Chapter: 3

Definition: The probability that \(X\)'s value is \(A\).
Page: 31
Chapter: 3
Variables: \(X\) a random variable and \(A\) a possible discrete value
Notation: \(Prob[X=A]\)

Name: Cumulative probability function
Definition: \(Prob[X\le x]\).
Page: 32
Chapter: 3
Variables: \(X\) a random variable whose domain is an ordered set, let \(x\) a possible value
Denoted by: \(F(x)\)

Name: Histogram
Definition: A graph, on horizontal label, the distinct possible value. On vertical label the probabilities.
Page: 34
Chapter: 3
Variables: \(X\) a random variable.

Name: The probability density function of \(X\)
Definition: A function \(f\) such that [$]Prob[x_1<X\le x_2]=\int_{x_1}^{x_2}f(x)\dd x[/$] for each \(x_1\le x_2\).
Page: 35
Chapter: 3
Variables: \(X\) a random variable taking real values.

Page: 37, 58 75
Name: The uniform distribution variable
Chapter: 3
Density function: The variable whose density function is \(\frac1{b-a}\) on \([a,b]\) and 0 elsewhere.
Mean: \(\frac {b+a}2\)
Variance: \(\frac 1{12}(b-a)^2\)
Skewness: 0
Kurtosis: \(\frac 9{15}\)
Moment generating function: \(1\) for \(t=0\), \(\frac{e^{tb}-e^{ta} }{t(b-a)}\) otherwise
Characteristic function: \(\frac{e^{itb}-e^{ita} }{it(b-a)}\)
Variables: \(a\le b\in\mathbb R\)
Intuition: Any value between \(a\) and \(b\) with the same probability
Mode: \((a,b)\)
Cumulative function: \(0\) for \(x\),
\(\frac{x-a}{b-a}\) for \(x\in[a,b)\),
\(1\) for \(x\ge b\)
Moment generating function computation: [$]\int_a^b\frac{e^tx}{b-a}\dd x[/$]. If \(t\ne 0\)
\(\left[\frac{e^{tx}}{t(b-a)}\right]_a^b\)=
\(\frac{e^{bt}-e^{at}}{t(b-a)}\).
For \(t=0\): [$]\int_a^b\frac{1}{b-a}\dd x=\left[\frac{x}{b-a}\right]_a^b=\frac{b-a}{b-a}=1[/$]
Characteristic function computation: [$]\int_a^b\frac{e^itx}{b-a}\dd x[/$]. If \(t\ne 0\)
\(\left[\frac{e^{itx}}{it(b-a)}\right]_a^b\) =
\(\frac{e^{ibt}-e^{iat}}{it(b-a)}\).
For \(t=0\): [$]\int_a^b\frac{1}{b-a}\dd x=\left[\frac{x}{b-a}\right]_a^b=\frac{b-a}{b-a}=1[/$]
Nth moment: \(\frac{1}{n+1}\sum_{k=0}^{n}a^kb^{n-k}\)
Mean computation: [$]\int_a^b\frac{x}{b-a}\dd x[/$] =
\(\left[\frac{x^2}{2(b-a)}\right]^b_a\) =
\(\frac{b^2-a^2}{2(b-a)}\) =
\(\frac{(b-a)(b+a)}{2(b-a)}\) =
\(\frac{b+a}{2}\)
Variance computation: [$]\int_a^b\frac{x^2}{b-a}\dd x-\left(\frac{b+a}{2}\right)^2[/$] =
\(\left[\frac{x^3}{3(b-a)}\right]^b_a-\frac{(b+a)^2}{4}\) =
\(\frac{b^3-a^3}{3(b-a)}-\frac{(b+a)^2}{4}\) =
\(\frac{b^2+ab+a^2}{3}-\frac{a^2+2ab+b^2}{4}\) =
\(\frac{4(b^2+ab+a^2)-3(a^2+2ab+b^2)}{12}\) =
\(\frac{b^2-2ab+a^2}{12}\) =
\(\frac{(b-a)^2}{12}\)
Cumulative function computation: [$]\int_a^t\frac{1}{b-a}\dd x[/$] =
\(\left[\frac{x}{b-a}\right]^t_a\) =
\(\frac{t-a}{b-a}\)
Nth cumulant: \(\frac{B_n}{n}\) with \(B_n\) the \(n\)-th Bernouilli number
Skewness computation: 3th central moment is 0, thus 0.
Kurtosis computation: \(\frac{\frac{(a-b)^4}{2^{4}5}}{\left(\frac{(b-a)^2}{12}\right)^2}\) =\(\frac{(a-b)^412^2}{2^{4}5(b-a)^4}\) =
\(\frac{12^2}{2^{4}5}\) =
\(\frac{9}{5}\)
Nth central moment: \(\left(\frac{a-b}{2}\right)^n\frac1{(n+1)}\) for \(n\) even
\(0\) otherwise
Nth moment computation: [$]\int_a^b\frac{x^n}{b-a}\dd x[/$] =
\(\left[\frac{x^{n+1}}{(n+1)(b-a)}\right]_a^b\)=
\(\frac{b^{n+1}-a^{n+1}}{(n+1)(b-a)}\)=
\(\frac{\sum_{k=0}^na^kb^{n-k}}{n+1}\)
Nth central moment computation: [$]\int_a^b\left(x-\frac{a+b}2\right)^n\dd x[/$] =
[$]\int_a^b\frac{(2x-a-b)^n}{2^n}\dd x[/$] =
\(\left[\frac{(2x-a-b)^{n+1}}{2^{n+1}(n+1)}\right]_a^b\) =
\(\frac{(b-a)^{n+1}+(a-b)^{n+1}}{2^{n+1}(n+1)}\) =

Name: Marginal distribution
Definition: The probability distribution over one variable when the other is fixed.
Page: 39
Chapter: 3
Variables: \(p(x,y)\) a probability distribution with two parameters,

Name: The probability function obtained from \(S\)
Definition: Let \(n\) the number of observation. Given a value \(v\) observed \(n_v\) times, \(p(X=v)=\frac{n_v}{n}\).
Page: 45
Chapter: 4
Variables: \(S\) a set of observation of real values

Name: The means of \(X\)
Definition: [$]\int X(x)f(x)\dd x[/$].
Page: 47, 68
Chapter: 4
Variables: \(X\) a random variable taking real values with distribution function \(f\)
Notation: \(E(x)\)
Name2: Expected value of \(X\)

Name: The median of \(F\)
Definition: A value \(V\) such that \(F[V]=\frac12\).
Page: 47
Chapter: 4
Variables: \(F\) a cumulative probability function

Name: The median of \(f\)
Definition: A value \(V\) such that [$]\int_{-\infty}^Vf(x)\dd x=\frac 12 [/$].
Page: 47
Chapter: 4
Variables: \(f\) a density probability function

Name: The mode of \(p\)
Definition: The solution of \(\frac{df(x)}{dx}=0\).
Page: 49
Chapter: 4
Variables: \(p\) a continuous density function
Assuming: A single local maxim

Smaller0: \(p\) is symmetrical
Greater0: mean, mode and median are equals
Page: 50
Chapter: 4
Variables: \(p\) a continuous density function

Smaller0: mode-median
Greater0: median-mean
Page: 50
Chapter: 4
Variables: \(p\) a continuous density function
Smaller to greater: equals twice
Assuming: Relation usually respected by mode, median and mean distance
Greater to smaller: equals half

Name: Means
Page: 53
344
Section: 1A
Chapter: 4,
27 Fremlin measure theory
Extra: ,
b
Closure: invariant under
Under: linear transformations
Not2: not
Under2: monotonic transformation
Under3: addition of distinct variables
Other fields (not shown on cards):
Prefix: is

Name: Median
Page: 53, 54
Chapter: 4
Closure: invariant under
Under: linear transformations
Under2: monotonic transformation
Not3: not
Under3: addition of distinct variables
Other fields (not shown on cards):
Prefix: is

Name: The mean deviation of \(f\)
Definition: Given \(\mu\) the mean, [$]\int |x-\mu|f(x)\dd x[/$].
Page: 54
Chapter: 4
Variables: \(f\) a probability density function

Name: First and third quartile
Definition: The values \(F^{-1}(\frac 14)\) and \(F^{-1}(\frac 34)\).
Page: 55
Chapter: 4
Variables: \(F\) a cumulative probability function

Name: The \(n\)-th moment about the mean
Definition: \(E((x-\mu)^n)\).
Page: 57
Chapter: 4
Variables: \(f\) a density probability function
Notation: \(\mu_n\)
Extra: Note that there is no absolute function in the expression

Smaller0: \(n\) is even
Greater0: nonnegative
Page: 58
Chapter: 4
Variables: \(f\) a probability distribution function and \(n>0\)
Hide middle: yes
Prefix Greater: the \(n\)-th moment is

Name: Variance
Definition: The 2nd moment.
Page: 57
Chapter: 4

Question: A probability distribution with multiple maximum is
Answer: probably composed of many unimodal distribution.
Page: 59
Chapter: 4

Name: Standard deviation
Definition: Square root of the variance.
Page: 57
Chapter: 4
Notation: \(\sigma\)

Name: Skewness
Definition: \[\frac{\mu _3}{\mu_2^{\frac 32} }\].
Page: 62
Chapter: 4
Variables: \(f\) a probability density function
Definition2: \(\frac{\mu_3}{\sigma^3}\)

Question: It is left skew/symmetric/right skew if
Answer: its skewness is positive 0 or negative.
Page: 62
Chapter: 4
Variables: \(f\) a probability density function

Name: The \(n\)-th moment about the origin
Definition: \(E(x^n)\)
Page: 62
Chapter: 4
Variables: \(f\) a probability density function

Name: Kurtosis
Definition: \[\frac{\mu_4}{\sigma^4}\]
Page: 64
Chapter: 4
Variables: \(f\) a probability density function
Typ: Real at least \(1\)

Name: Distribution with kurtosis 1
Page: 64
Chapter: 4
Example: A variable with two values at probability one half

Name: Distribution with infinite kurtosis
Page: 64
Chapter: 4
Example: \(t\) distribution with four degree of freedom

Name: Leptokurtic distribution
Definition: A distribution with kurtosis greater than 3.
Page: 65
Chapter: 4
Extra: Like two kangaroos, known for lepping

Name: Platykurtic distribution
Definition: A distribution with kurtosis smaller than 3.
Page: 65
Chapter: 4
Extra: Like a plapytus

Question: \(E(a+bX)=\)
Answer: \(a+bE(X)\).
Page: 69
Chapter: 5
Variables: \(X\) a random variable, \(a,b\in\mathbb R\)

Question: \(V(a+bX)=\)
Answer: \(b^2V(X)\).
Page: 69
Chapter: 5
Variables: \(X\) a random variable
\(a,b\in\mathbb R\)

Question: \(E(X+Y)=\)
Answer: \(E(X)+E(Y)\).
Page: 70
Chapter: 5
Variables: \(X,Y\) random variables

Name: The \(r\)-th moment about the origin
Definition: \(E(X^r)\).
Page: 71, 86
Chapter: 5, 6
Variables: \(X\) a random variable
\(r\in\mathbb N\)
Notation: \(\mu'_r\)
Definition2: \(M_X^{(r)}(0)\)

Question: Relation between the two second moments:
Answer: \(\mu_2=\mu'_2-\mu^2\).
Page: 71
Chapter: 5
Variables: \(X\) a random variable

Smaller0: \(E(XY)\)
Greater0: \(E(X)E(Y)\)
Page: 71
Chapter: 5
Variables: \(X,Y\) random variables
Smaller to greater: equals
Assuming: \(X\), \(Y\) are independant

Q: \(V([...])=[...]\).
A: X+Y; V(X)+V(Y)
Page: 71
Chapter: 5
Variables: \(X,Y\) random variables
Assuming: \(X\), \(Y\) are independant

Name: The covariance of \(X\) and \(Y\)
Definition: \(E[(X-\xi)(Y-\eta)]\) with \(\xi\) and \(\eta\) their means.
Page: 73
Chapter: 5
Variables: \(X,Y\) random variables
Abbreviation: \(\text{Cov}(X,Y)\)

Question: \(V(X+Y)=\)
Answer: \(V(X)+V(Y)+2\text{Cov}(X,Y)\).
Page: 73
Chapter: 5
Variables: \(X,Y\) random variables

Q: Intuitively, covariance is [...] when [...].
A: negative; the two variables have opposite change from the mean
Page: 74
Chapter: 5

Name: The correlation coefficient
Definition: \(\frac{Cov(X,Y)}{\sqrt{V(X)V(Y)} }\).
Page: 74
Chapter: 5
Variables: \(X,Y\) two variables
Typ: A real between \(-1\) and \(1\)
Denoted by: \(\rho\)

Q: The correlation coefficient of two variables is [...] if [...].
A: -1,1 respectively; the variables are on a straight line with negative/positive slope respectively
Page: 74
Chapter: 5

Q: \(M_{X_1[...]X_2}(t)=M_{X_1}[...]M_{X_2}\).
A: +; \cdot
Page: 76
Chapter: 5
Variables: \(X_1,X_2\) random variables
Assuming: The variables are independant

Smaller0: \(M_{a+bX}(t)\)
Greater0: \(e^{at}M_X(bt)\)
Page: 77
Chapter: 5
Variables: \(X\) a random variable,
\(a,b\in\mathbb R\)
Smaller to greater: =
Hide middle: yes

Name: Characteristic function of \(X\)
Definition: \(E(e^{itX})\)
Page: 77
Chapter: 5
Variables: \(X\) a random variable
Context: Random variable
Notation: \(\phi_{X}(t)\)
Extra: mgf iX

Name: positive covariance
Page: 74
Chapter: 5
Intuition: the two variables increases or decreases together.

Name: A random variable
Definition: A measurable function, whose domain is a space of measure one.
Page: Not in the book
Chapter: 3

Page: 83
Name: The probability of \(x\) sucess in \(n\) try, in a particular order
Chapter: 6
Density function: Sending \(0\le x\le n\) to \(P^x(1-P)^{n-x}\) and everything else to 0
Variables: \(P\) a probability of success
\(0\le x\le n\)

Q: [...] is [...].
A: The probability of \(x\) sucess in \(n\) try, in any order; \(P^x(1-P)^{n-x}{n\choose x}\)
Page: 84
Chapter: 6
Variables: \(P\) a probability of success
\(0\le x\le n\)

Page: 86
Name: The proportion of success in the binomial distribution with parameters \(P\) and \(n\)
Chapter: 6
Density function: \(P^x(1-P)^{n-x}/n\) for \(0\le x\le n\)
Mean: \(nP\)
Variance: \(P(1-P)/n\)

Name: An output wire of \(u\)
Definition: A gate \((v,u)\in E\).
Page: 8
Chapter: 1
Section: 2
Variables: \(B\) a basis,
\(C=(V,E,\alpha,\beta,\omega)\) a circuit and
\(u\in V\).
Kind: Def

Page: 90, 91
Name: The poisson Distribution
Chapter: 6
Density function: [$]\defFun{\mathbb N}{[0,1]}{n}{\frac{e^{-\mu}\mu^n}{n!} }[/$]
Mean: \(\mu\)
Skewness: \(\frac1{\sqrt\mu}\)
Kurtosis: \(3+\frac1\mu\)
Moment generating function: \(e^{\mu(e^t-1)}\)
Characteristic function: \(e^{\mu(e^{it}-1)}\)
Variables: \(\mu\ge0\)
Intuition: The number of occurence of one event, assuming all event are independant, and happen with median value \(\mu\)
Moment generating function computation: \(\sum_{k=0}^\infty e^{tk}\frac{e^{-\mu}\mu^k}{k}\) =
\(e^{-\mu}\sum_{k=0}^\infty\frac{(\mu e^t)^k}{k!}\) =
\(e^{-\mu}e^{\mu e^t}\)
Characteristic function computation: \(\sum_{k=0}^\infty e^{ktn}\frac{e^{-\mu}\mu^k}{k!}\) =
\(e^{-\mu}\sum_{k=0}^\infty\frac{(\mu e^{it})^k}{k!}\) =
\(e^{-\mu}e^{\mu e^{it}}\)
Standard deviation: \(\sqrt\mu\)
Nth moment: [$]\sum_{k=0}^n\mu^k{n\brace m}[/$]
Mean computation: \(\sum_{k=0}^\infty\frac{ne^{-\mu}\mu^k}{k!}\)=
\(\mu\sum_{k=1}^\infty\frac{e^{-\mu}\mu^{n-1}}{(n-1)!}\)=
\(\mu\sum_{k=0}^\infty\frac{e^{-\mu}\mu^k}{k!}\)=
\(\mu e^{-\mu}\sum_{k=0}^\infty\frac{k\mu^k}{k!}\)=
\(\mu e^{-\mu}e^\mu\) = \(\mu\)
Variance computation: \(\sum_{k=0}^\infty\frac{k^2\mu^k}{k!}-\mu^2\)=\(e^{-\mu}\sum_{k=0}^\infty\frac{((k-1)+1) \mu^{k}}{(k-1)!}-\mu^2\)=
\(e^{-\mu}\left(\sum_{k=0}^\infty\frac{(k-1) \mu^{k}}{(k-1)!}+\sum_{k=0}^\infty\frac{\mu^{k}}{(k-1)!}\right)-\mu^2\) =
\(e^{-\mu}\left(\mu^2\sum_{k=0}^\infty\frac{\mu^{k-2}}{(k-2)!}+\mu\sum_{k=0}^\infty\frac{\mu^{k}}{(k-1)!}\right)-\mu^2\) =
\(e^{-\mu}\left(\mu^2e^\mu+\mu e^\mu\right)-\mu^2\) =
\(\mu^2+\mu -\mu^2\) = \(\mu\)
Nth factorial moment: \(\mu^n\)

Smaller0: \((X+Y)\)
Greater0: the Poisson distribution with parameter \(\mu+\nu\)
Page: 91
Chapter: 6
Variables: \(X,Y\) Poisson distributions with parameter \(\mu,\nu\)
Smaller to greater: equals
Hide middle: yes
Assuming: \(Y,X\) independant

Name: A standard normal variate
Definition: A normal variable with mean \(0\) and variance \(1\).
Page: 112
Chapter: 7

Q: [...] is [...].
A: Their sum; the normal variable with means \(\mu_0+\mu_1\) and variance \(\sigma_0^2+\sigma_1^2\)
Page: 113
Chapter: 7
Variables: \(X_0,X_1\) normal variables with means \(\mu_i\) and variance \(\sigma_i^2\)

Page: 97
Name: The exponential distribution
Chapter: 6
Density function: \(\lambda e^{-\lambda x}\)
Mean: \(\frac 1\lambda\)
Variance: \(\frac1{\lambda^2}\)
Skewness: 2
Kurtosis: 9
Moment generating function: \(t\mapsto \frac{\lambda}{\lambda-t}\)
Characteristic function: \(t\mapsto \frac{\lambda}{\lambda-it}\)
Variables: \(\lambda>0\)
Intuition: Describes the time between events in a Poisson process.
Mode: 0
Cumulative function: \(1-e^{-\lambda x}\)
Median: \(\frac{\log 2}{\lambda}\)
Moment generating function computation: [$]\lambda\int_0^\infty e^{tx}e^{-\lambda x}\dd x[/$] =
[$]\lambda\int_0^\infty e^{-(\lambda -t)x}\dd x[/$] =
\lambda\(\left[\frac{e^{-(\lambda-t)x}}{-(\lambda-t)}\right]_0^\infty\) =
\(\frac{\lambda}{\lambda-t}\)
Characteristic function computation: [$]\lambda\int_0^\infty e^{itx}e^{-\lambda x}\dd x[/$] =
[$]\lambda\int_0^\infty e^{-(\lambda -it)x}\dd x[/$] =
\(\left[\frac{e^{-(\lambda-it)x}}{-(\lambda-it)}\right]_0^\infty\) =
\(\frac{\lambda}{\lambda-it}\)
Standard deviation: \(\frac1{\lambda}\)
Mean computation: [$]\int_0^\infty x\lambda e^{-\lambda x}\dd x[/$] =
[$]\frac1\lambda\int_0^\infty y e^{-y}\dd y[/$] =
\(\frac1\lambda\left[-(y+1)^e{-y}\right]_0^\infty\) =
\(\frac1\lambda\)
Cumulative function computation: [$]\int_0^t\lambda e^{-\lambda x}\dd x[/$] =
\(\left[-e^{-\lambda x}\right]_0^t\)=
\(-e^{-\lambda t}+e^{-\lambda 0}\)=
\(1-e^{-\lambda t}\)
Extra: \(M(t)=\frac{\lambda}{\lambda-t}\)
\(M'(t)=\frac{\lambda}{(\lambda-t)^2}\)
\(M''(t)=\frac{-\lambda(2t-2\lambda)}{(\lambda-t)^4}\)
\(M''(0)=2/\lambda^3\)

Name: A mesokurtic distributiton
Definition: A distribution probability whose kurtosis is \(3\).
Page: wp

Smaller0: \(\sum\mu_i\)
Greater0: \(\mu\)
Page: 115
Chapter: 7
Section: Central limite theorem
Variables: \((X_i)\) independent random variables, with mean \(\mu_i\), variance \(\sigma^2_i\), \(j\)-th moment \(\mu_{j,i}\)
\(Y=\sum X_i\), with mean \(\mu\), variance \(\sigma^2\), \(j\)-th moment \(\mu_{j}\)
Smaller to greater: =

Smaller0: \(\sum\sigma^2_i\)
Greater0: \(\sigma^2\)
Page: 115
Chapter: 7
Section: Central limite theorem
Variables: \((X_i)\) independent random variables, with mean \(\mu_i\), variance \(\sigma^2_i\), \(j\)-th moment \(\mu_{j,i}\)
\(Y=\sum X_i\), with mean \(\mu\), variance \(\sigma^2\), \(j\)-th moment \(\mu_{j}\)
Smaller to greater: =

Smaller0: \(\sum\mu_i\)
Greater0: \(\mu_{j,i}\)
Page: 115
Chapter: 7
Section: Central limite theorem
Variables: \((X_i)\) independent random variables, with mean \(\mu_i\), variance \(\sigma^2_i\), \(j\)-th moment \(\mu_{j,i}\)
\(Y=\sum X_i\), with mean \(\mu\), variance \(\sigma^2\), \(j\)-th moment \(\mu_{j}\)
Smaller to greater: =

Smaller0: \(M_{Y-\mu}\)
Greater0: \(\Pi M_{X_i-\mu}(t)\)
Page: 115
Chapter: 7
Section: Central limit theorem
Variables: \((X_i)\) independent random variables, with mean \(\mu_i\), variance \(\sigma^2_i\), \(j\)-th moment \(\mu_{j,i}\)
\(Y=\sum X_i\), with mean \(\mu\), variance \(\sigma^2\), \(j\)-th moment \(\mu_{j}\)
Smaller to greater: =

Definition: The m.g.f. of the standardised variate \(\frac{X-\mu}\sigma\)
Page: 115
Chapter: 7
Section: Central limite theorem
Variables: \(X\) a random variable
Notation: \(M^*(t)\)

Page: 124, wp
Name: The \(\Xi^2\) distribution with \(f\) degrees of freedom
\(\Xi^2_{[f]}\) chi-square(d) distribution
Chapter: 8
Density function: Equivalently:
-\(\frac{x^{\frac f2-1}e^{-\frac f2}}{2^{f/2}\Gamma(f/2)}\)
-\(\frac{x^{\frac f2-1}e^{-\frac f2}}{\Pi_{i=1}^{\frac{f-1}2}(2i-1)\sqrt{2\pi}}\) for \(f\) odd
-\(\frac{x^{\frac f2-1}e^{-\frac f2}}{2\Pi_{i=1}^{\frac{f-1}2}(2i)}=\frac{x^{\frac f2-1}e^{-\frac f2}}{2^{\frac f2}(\frac f2-1)!}\) for \(f\) even
Mean: \(f\)
Variance: \(2f\)
Skewness: \(\sqrt{\frac 8f}\)
Kurtosis: \(\frac{12}f\)
Moment generating function: \((1-2t)^{-\frac f2}\) for \(t<\frac12\)
Characteristic function: \((1-2it)^{-\frac f2}\)
Variables: \(f\in\mathbb N\)
Intuition: Sum of the squares of \(f\) independants standard normal variates
Mode: \(\text{max}(k-2,0)\)
Cumulative function: \(\frac{\gamma(\frac f2,\frac x2)}{\Gamma(f/2)}=P\left(\frac f2,\frac x2\right)\)
Median: \(\sim f\left(1-\frac2{9f}\right)^3\)
Section: The \(\Xi^2\), \(t\) and \(F\) distributions

Page: 124
Name: \(\Xi^2_{[1]}\)
Chapter: 8
Density function: \(\frac{x^{-\frac 12}e^{-\frac 12}}{2^{\frac 12}\Gamma(\frac 12)}\)
\(\frac{e^{x^2}}{2\pi}\)
Mean: \(1\)
Variance: \(2\)
Skewness: \(2\sqrt{2}\)
Kurtosis: \({12}\)
Moment generating function: \((1-2t)^{-\frac 12}\) for \(t{\lt}\frac12\)
Characteristic function: \((1-2it)^{-\frac 12}\)
Intuition: Square of a standard normal variates
Mode: \(0\)
Cumulative function: \(\frac{\gamma(\frac 12,\frac 12)}{\Gamma(\frac 12)}=P\left(\frac 12,\frac 12\right)\)
Section: The \(\Xi^2\), \(t\) and \(F\) distributions

Question: The sum of a chi-squared distribution with \(f_1\) and \(f_2\) degrees of variance is
Answer: the chi-squared distribution with \(f_1+f_2\) degrees of variance.
Page: 126
Chapter: 8
Section: The \(\Xi^2\), \(t\) and \(F\) distributions

Smaller0: \(p\) is skew with long tail on the right
Greater0: mode
Page: 50
Chapter: 4
Variables: \(p\) a continuous density function
Greater1: median
Connect Greater0: <
Greater2: mean

Smaller0: \(P(A\text{ or }B)\)
Greater0: \(P(A)+P(B)-P(A\text{ and }B)\)
Page: 14
Chapter: 2
Variables: \(A,B\) two events.
Smaller to greater: =
Name: The Law of Addition

Smaller0: \(X\) and \(Y\) are independant
Greater0: 0
Page: 73
Chapter: 5
Variables: \(X,Y\) random variables
Hide middle: yes
Prefix Greater: the covariance of \(X\) and \(Y\) is