Pre-knowledge:
\ (1 \) high school mathematics knowledge.
\ (2 \) Higher Mathematics (differential, definite integral, indefinite integral, Taylor expansion, limits, etc.)
- Definite integral used in the calculation mode: Newton - Leibniz formula :( \ (F. () \) Of \ (F () \) of the original function, i.e. \ (F ^ { '} ( ) = f () \) )
\[ \int_a^b{f(x)dx}=F(b)-F(a) \]
- Taylor value theorem \ (. 1 \) : \ (F (X) = F (x_0) + F '(x_0) (X-x_0) + \ FRAC {F' '(x_0) 2} {!} (X- x_0) ^ 2 + ... + \ FRAC {F ^ {(n-)} (x_0) n-} {!} (X-x_0) ^ n-R_n + (X) \) , satisfies \ (f (x) \ ) in \ (x_0 \) at there \ \) (n- order derivative, \ (X \) is the \ (x_0 \) any value of a neighborhood, \ (R_n (X) = O ((X-x_0 )) ^ n \) called Pei Yanuo remainder.
- Taylor value theorem \ (2 \) : \ (F (X) = F (x_0) + F '(x_0) (X-x_0) + \ FRAC {F' '(x_0) 2} {!} (X- x_0) ^ 2 + ... + \ FRAC {F ^ {(n-)} (x_0) n-} {!} (X-x_0) ^ n-R_n + (X) \) , satisfies \ (f (x) \ ) in \ (x_0 \) of the neighborhood have a \ (n + 1) \ -order derivative, \ (X \) is the \ (x_0 \) any value of the neighborhood, \ (R_n (X) = \ frac {f ^ {n + 1} (\ xi)} {(n + 1)!} (x-x_0) ^ {n + 1} \) called Lagrange remainder \ ((\ XI \ ) in \ (x_0 \) and \ (X \) between \ () \) .
- Take: Maclaurin formula \ (x_0 = 0, \ xi = \ theta x (0 <\ theta <1) \) Taylor expansion upon.
- Common Maclaurin formula ( important )
- \(e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+\frac{e^{\theta x}}{(n+1)!}x^{n+1}\)
- \(sinx=x-\frac{x^3}{3!}+\frac{x^5}{5!}-...+(-1)^{m-1}\frac{x^{2m-1}}{(2m-1)!}+R_{2m}(x)\)
\ (3 \) Mean Value Theorem (Rolle, Lagrange, Cauchy mean value) (understanding of the linkages slope)
- Rolle's theorem: \ (F (X) \) satisfies \ ([a, b] \ ) continuous, \ ((A, B) \) can be turned on, \ (F (A) = F (B) \ ) the \ (\ exists \ xi (a <\ xi <b) \) have \ (f '(\ xi) = 0 \)
- Lagrange Theorem: \ (F (X) \) satisfies \ ([a, b] \ ) continuous, \ ((A, B) \) on a guide, the \ (\ exists \ xi ( a <\ xi <b) \ ) have \ (f (b) -f ( a) = f '(\ xi) (ba) \)
- Mean Value Theorem: \ (F (X), F. (X) \) satisfies \ ([a, b] \ ) continuous, \ ((A, B) \) on a guide, \ (\ FORALL X \ in (a, b) F '( x) \ ne0 \) then \ (\ exists \ xi (a <\ xi <b) \) have \ (\ frac {f (a ) -f (b)} {F ( a) -F (b)} = \ frac {f '(\ xi)} {F' (\ xi)} \)
Chapter I: Basic Concepts (or a little of it, really is high school knowledge ......)
Chapter Two: random variables and their distribution
\ (1 \) In general, random variables into discrete and continuous, discrete number of columns can be used analogy, the continuous function may be used analogy.
Note that, in the definite integral continuous discrete equivalent of \ (\ Sigma \) .
\ (2. \) \ (P \ {X-\} \) denotes a random variable \ (X-\) probability, the following are a few important distribution
- \ ((0-1) \) Distribution: \ (P \ {= X-K \ K ^ P} = (. 1-P) ^ K, K = 0,1, (0 <P <. 1) \)
- Bernoulli experiment: The results in both cases only occur or not occur.
- \ (n \) heavy Bernoulli experiment: do \ (n \) Bernoulli experiments, the number of incidents.
Binomial Distribution: \ (P \ {X-K = \ n-} = {\} the Choose K ^ P K (. 1-P) NK ^ {}, K = 0,1,2,3 ... \) Us random variable \ (X-\) subject to parameters \ (n, p \) binomial, referred to as \ (X \ sim b (n , p) \)
Poisson distribution: \ (P \ {= X-K \ =} \ {FRAC \ ^ KE the lambda ^ {-}} {K K!}, K = 0,1,2 ..., \ the lambda> 0 \) we call random variables \ (X \) subject to parameters \ (\ lambda \) Poisson distribution, denoted by \ (X \ sim \ pi ( \ lambda) \)
About Poisson distribution, binomial distribution may be utilized to establish contact with the Poisson's Theorem, when \ (np_n = \ lambda \) i.e. when there
\ [\ lim_ {n \ to \ infty} {n \ choose k} p ^ k_n ( 1-p_n) ^ {nk} = \ frac {\ lambda ^ ke ^ {- k}} {k!} \]
\ (3 \) distribution function \ (F. (X) \) , in the case of one-dimensional, and look like as a prefix, that is: \ (F. (X) = P \ {X-\ Leq X \} , F (- \ infty) = 0, F (\ infty) = 1 \)
\ (4 \) of the product at the distribution function, there is the probability density \ (F (X) \) , namely: \ (F. (X) = \ X ^ int _ {- \ infty} F (T) dt, P \ {x_1 <X \ leq x_2 \} = \ int ^ {x_2} _ {x_1} f (x) dx \)
Uniform: the random variable \ (X-\) probability density
\ [f (x) = \ left \ {\ begin {array} {ll} \ frac {1} {ba} & a <x <b \\ 0 & x \ leq a || x \ geq b \
end {array} \ right. \] we call \ (X-\) in the section \ () \ (a, b ) subject to the uniform distribution, referred to as \ (X ~ U (a , b) \) .Exponential distribution: random variable \ (X-\) probability density \ ((\ Theta> 0) \)
\ [F (X) = \ left \ {\ the begin {Array} {LL} \ FRAC {. 1} {\ Theta } e ^ {- \ frac {
x}. {\ theta}} & x> 0 \\ 0 & x \ leq 0 \ end {array} \ right \] we call \ (X-\) in the section \ ((a, b) \) subject to parameters for the \ (\ theta \) exponential distribution.Normal (this stuff and \ (e ^ x \) has the similarities between, with the heart): random variable \ (X-\) probability density \ (\ mu, \ sigma \ ) is a constant \ ((\ Sigma > 0) \)
\ [F (X) = \ {FRAC. 1} {\ sqrt {2 \} PI \ Sigma ^ {E} - \ FRAC {(X-\ MU) ^ 2} {2 \ Sigma ^ 2 }}, - \ infty <x
<\ infty \] we call \ (X-\) in the section \ ((a, b) \ ) on the subject to parameters \ (\ mu, \ sigma \ ) of the normal distribution, denoted is \ (X-\ SIM N (\ MU, \ Sigma ^ 2) \) .Normal \ (f (x) \) image with the highest point \ (x = \ mu \) at, \ (\ Sigma \) to determine its shape.
\ (\ mu = 0, \ sigma = 1 \) we said standard normal random variable, i.e.
\ [\ varphi (x) = \ frac {1} {\ sqrt {2} \ sigma} e ^ { - \ frac {x ^ 2} {2}}, - \ infty <x <\ infty \]If \ (X \ sim N (\ mu, \ sigma ^ 2) \) then \ (the Z = \ {X-FRAC \ MU} {\ Sigma} \ SIM N (0,1) \) , applied this Jiacha table, we can find any normal probability density.
Chapter III multi-dimensional random variables and their distribution
\ (1 \) First, the first focus, we must learn to multi-dimensional partial derivatives, seen as one-dimensional constant, to be another derivative.
\ (2. \) \ (F. (X, Y) \) (the joint probability distribution becomes) similarly defined above, with the area and can be appreciated, \ (F (X, Y) = \ {FRAC \ partial ^ 2 F. (X, Y)} {\ partial X \ partial Y} \) , seeking method: firstly \ (F (x, y) \) a \ (X \) derivative, and then the result is \ (Y \) derivation.
\ (3 \) marginal probability density: \ (F_X (X) = \ int ^ \ infty _ {- \ infty} F (X, Y) Dy, f_y (X) = \ int ^ \ infty _ {- \ infty} F (X, Y) DX \) .
For a \ (F (X) \) , if its \ ([- a, a] \) integrable on there:
\ [\ int ^ A _ {- A} F (X) DX = \ left \ { \ begin {array} {ll} 2 \ int ^ a_0f (x) dx & f (-x) = f (x) \\ 0 & f (-x) = - f (x) \ end {array} \ right \].Two-dimensional normal distribution:
联合分布:
\ [f (x) = \ frac {1} {2 \ pi \ sigma_1 \ sigma_2 \ sqrt {1- \ rho ^ 2}} exp \ {\ frac {-1} {2- (1- \ rho ^ 2)} [\ frac {(x- \ mu_1) ^ 2} {\ sigma_1} ^ 2 -2 \ rho \ frac {(x- \ mu_1) (y \ mu_2)} {\ sigma_1 \} + sigma_2 \ {frac (y \ mu_2) ^ 2} {\ sigma_2 ^ 2}] \} \]Marginal distribution to meet the one-dimensional normal distribution.
\(4.\)条件概率：\(P\{X=x_i|Y=y_i\}=\frac{P\{X=x_i,Y=y_i\}}{P\{Y=y_i\}}=\frac{p_{ij}}{p_{\cdot j}}\)
Conditional probability density: \ (X-F_ {|} the Y (x_y) = \ FRAC {F (X, Y)} {f_y (Y)} \) (in this formula may be employed to push down the joint distribution density)
Uniform distribution of: setting \ (\ G) a bounded region on the plane area, the area \ (A \) , \ ((X-, the Y) \) having a probability density:
\ [F (X, Y) = \ left \ {\ begin {array} {
ll} \ frac {1} {A} & (x, y) \ in G \\ 0 & (x, y) \ notin G \ end {array} \ right. \] us \ ((X, Y) \ ) in \ (G \) a uniform distribution (which is not necessarily uniformly distributed edge distribution)
\ (5 \) independent random variables: The method of determination: \ (P \ {X-\ Leq X, the Y \ Y Leq \} = P \ {X-\ Leq X \} P \ {the Y \ Y Leq \} , f (x, y) = f_X (x) f_Y (y) \)
Chapter 4: numerical characteristics of random variables:
\ (1 \) expectation (which is essentially a weighted average): \ (E (X-) = \ int ^ \ infty _ {- \ infty} XF (X) DX \)
Binomial distribution: \ (E (the X-) = NP \)
Normal distribution: \ (E (the X-) = \ MU \)
Uniform distribution of: \ (E (X-) = \ FRAC {A + B} {2} \)
Poisson distribution: \ (E (the X-) = \ the lambda \) (proved once again used a Maclaurin formula)
Exponential distribution: \ (E (the X-) = \ Theta \) ( proof ) (Note: the proof of \ (\ lambda \) and here \ (\ theta \) are reciprocal, write here \ (\ theta \) and the book just for consistency)
Theorem: If the random variable \ (X-\) is the probability density \ (F (X) \) , \ (\ int ^ \ infty _ {- \ infty} G (X) F (X) DX \) absolute converged, there are:
\ [E (the Y) = E (G (X-)) = \ int ^ \ infty _ {- \ infty} G (X) F (X) DX \]Note: \ (E (the X-) \) is a constant. \ (E (CX) = CE (X), E (X + Y) = E (X) + E (Y), E (XY) = E (X) E (Y) \)
\ (2 \) variance: \ (D (X) = \ int ^ \ infty _ {- \ infty} [xE (X-)] ^. 2F (X) DX = E (X-^ 2) - [E (X-) ] ^ 2 \)
Random variable \ (X-\) has a mathematical expectation \ (E (X) = \ mu, D (x) = \ sigma ^ 2 \) then \ (X ^ * = \ frac {X- \ mu} {\ sigma} \) becomes \ (X \) standardized variables.
- Normal: \ (D (X-) = \ Sigma ^ 2 \)
- Poisson distribution: \ (D (the X-) = \ the lambda \)
- Uniform distribution of: \ (D (X-) = \ {FRAC (BA) 12 is ^ {2}} \)
- Exponential distribution: \ (D (X-) = \ Theta ^ 2 \)
- Binomial Distribution: \ (D (X-) NP = (. 1-P) \)
- \(D(C)=0,D(CX)=C^2D(X),D(C+X)=D(X)\)
\ (D (X-+ the Y) = D (X-) + D (the Y) + 2E \ {(XE (X-)) (YE (the Y)) \} \) , particularly when \ (X, Y \) independently of each other when \ (D (X + Y) = D (X) + D (Y) \)
\ (3 \) covariance and correlation coefficient:
Covariance: \ (Cov (X-, the Y) = E \ {[XE (X-)] [YE (the Y)] \} \) , the correlation coefficient: \ (\ rho_ {the XY} = \ FRAC {Cov (X-, Y)} {\ sqrt {D (X) D (Y)}} \)
协方差性质: \ ((Ax, By) = abCov (X, Y), (X_1 + X_2, Y) = (X_1, Y) + (X_2, Y) \)
柯西施瓦茨inequality: \ (| Cov (the X-, the Y-) ^ 2 | \ Leq D (the X-) D (the Y-) \)
When \ (\ rho_ {XY} = 0 \) , the said \ (X, Y \) is not relevant. A two-dimensional normal distribution \ (\ Rho \) is \ (X, Y \) is the correlation coefficient \ (\ rho_ {XY} \ )