Articles in mathematic category

Positive Semi Definite

2019/2/7
| mathematic
PSD differential equation Any matrix $A_{d \times d}$ is said to be PSD if $x^TAx \geq$ 0 $\forall$ vector $x_{d \times 1}$. Examples Identity matrix $I_{d \times d}$: $$ A = \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & 1 & 0 & 0 \newline 0 & 0 & 1 & 0 \newline 0 & 0 & 0 & 1 \end{bmatrix} $$ Then \begin{align} x^TAx &= \sum_{i = 1}^d \sum_{i = 1}^d A_{i,j}x_ix_j \newline &= \sum_{i = 1}^d A_{i,i} x_i^2\newline &= \sum_{i = 1}^d x_i^2 \newline &\geq 0 \end{align}……

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Matrix

2019/2/1
| mathematic
Determinant Calculation Determinant of 2x2 matrix $$ A= \begin{bmatrix} a & b \newline c & d \end{bmatrix} $$ $|A| = det(A) = ad -bc$ Determinant of 3x3 matrix, also called expansion of the determinant by first row. Link. $$ B= \begin{bmatrix} a & b & c \newline d & e & f \newline g & h & k \end{bmatrix} $$ $|B| = det(B) = a\begin{vmatrix} e & f \newline h & k \end{vmatrix} -b\begin{vmatrix} d & f \newline g & k \end{vmatrix} +c\begin{vmatrix} d & e \newline g & h \end{vmatrix}$……

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Marginal Distribution

2019/1/31
| mathematic
Marginal Distribution: It’s a function that gives the probability based on only subset of the variables.1. For example, Or in mathematical way, for discrete $$Pr(X = x) = \sum_y Pr(X = x, Y = y) = \sum_y Pr(X = x | Y = y) Pr(Y = y)$$ and for continuous $$p_X(x) = \int_y p_{X,Y}(x,y) dy = \int_y p_{X|Y}(x|y)p_Y(y) dy$$ and it can also be written as Expected Vaue $$p_X(x) = \int_y p_{X|Y} (x|y) p_Y(y) dy = \mathbb{E}_Y[p_{X|Y} (x|y)]$$……

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Margin of Error

2019/1/27
| mathematic
Z-Score vs T-Score Z-Score Link Z-Score’s formula $$z = \frac{X - \mu}{\sigma}$$ where X = sample mean, $\mu$ = population means, $\sigma$ = population standard deviation. Also, we use Z Score when sample size >= 30 or we know the population’s mean dna SD. Z Table T-Score T-Score T-Score’s formula $$T = \frac{X - \mu}{s/ \sqrt{n}}$$ where X = sample mean, $\mu$ = population mean, s = sample standard deviation, and n = sample size.……

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Probability

2019/1/24
| mathematic
Discrete Random Variables A random variable is a number whose value depends upon the outcome of a random experiement. Such as tossing a coin 10 times and let X be the number of Head. A discrete random variable X has finitely countable values $x_i = 1, 2…$ and $p(x_i) = P(X = x_i)$ is called probability mass function. Probability mass functions has following properties: For all i, $p(x_i) > 0$ For any interval $P(X \in B) = \sum_{x_i \in B}p(x_i)$ $\sum_{i}p(x_i) = 1$ There are many types of discrete random variable……

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