Postgraduate Mathematics Note1—Division Framework

calculus

Calculus taught me why the area of ​​an ellipse = $\pi$ab.

Implicit function derivation Rule

How to understand Lagrange's function extremum?

All things can be integrated (all functions can find the original function?) - number is the universe

Linear Algebra

• It’s doubtless that Gitmind&Blog is best place for taking notes！

probability theory

1. What is the negative binomial distribution
2. The function of the three major distribution variables, mean, variance - the essence is calculation, the key point is memory (understanding)

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1. Chi-square distribution, originate from $\tau$ 函数——$t\left(x\right)={\int }_0^{+\infty }{e}^{-t}{t}^{x-1}dt$，prop1：$t(x+1)=x\tau \left(x\right)$
   produce$k_n(x)=\frac{e^{-x/2}{x}^{-(n-2)/2}}{\tau (n/2)2^{n/2}}$, recorded as $x_n^2$, satisfying the law of addition;

Proof: the probability density of the square of the normal distribution is the same as that of the first-order chi-square distribution , and the probability density of the normal distribution is the same as that of the chi-square distribution by induction in the calculation;

inclusion 1：Xi corresponds with index distribution，2$\lambda X_i$ complies with $x_2^2$

• Chi-square distribution followed by produce T distribution and F distribution

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4. The normal distribution is the sum of independent and identically distributed variables (and other statistics) at n $n\to +$ an approximation of$\infty$

Mathematical statistics

common pipeline：

1. Given variable distribution, known sample, estimate parameters (point estimation) or judge parameters (parametric test);
   bayes estimation obtains estimated value $\leftarrow$ sum of conditional probabilities = 1
2. The distribution of parameters and variables is known (or forced by large sample method/beyas), and the sample interval is estimated according to a given probability (interval estimation);
3. It is also possible to judge the confidence of the variable distribution (test of goodness of fit) when the parameters and samples are known.

• Visible 3 pivots = variable distribution, parameter, sample
• The three major distributions are born for the subsequent interval estimation and parameter testing

4. How to Bayesian estimate normal distribution variance?
   

Process of (specify level $\alpha$ gives a test of the hypothesis for the parameter):

• Given a sample and assumptions (distribution of variables)
• Type I Error = Parameters fit the distribution, but the sample deviates from the norm

1. Represents the first type of error probability (according to the given hypothesis H and the test with the ball ( $\overline{X}\ge C$ )
2. The identity is transformed into a sample function, using the classical conclusions, corresponding to the distribution functions of the three major statistics,
3. According to "with level $\alpha$ ” (for any satisfying H0), as long as the maximum value of the distribution function$\le \alpha$ , the parameter C in the solution test of the linear equation in one variable