Chi-Squared Distribution
Tags: #Math #StatisticsEquation
$$Q = \sum^{k}_{i=1} Z^{2}_{i} \sim \chi^{2}(k), \\ f(k) = \frac{1}{2^{k/2} \Gamma (k/2)} x^{k/2-1} e^{-x/2}, \\ F(k) = \frac{1}{\Gamma (k/2)} \gamma (\frac{k}{2},\frac{x}{2})$$Latex Code
Q = \sum^{k}_{i=1} Z^{2}_{i} \sim \chi^{2}(k), \\
f(k) = \frac{1}{2^{k/2} \Gamma (k/2)} x^{k/2-1} e^{-x/2}, \\
F(k) = \frac{1}{\Gamma (k/2)} \gamma (\frac{k}{2},\frac{x}{2})
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Introduction
Equation
$$Q = \sum^{k}_{i=1} Z^{2}_{i} \sim \chi^{2}(k)$$ $$f(k) = \frac{1}{2^{k/2} \Gamma (k/2)} x^{k/2-1} e^{-x/2} $$ $$F(k) = \frac{1}{\Gamma (k/2)} \gamma (\frac{k}{2},\frac{x}{2}) $$
Latex Code
Q = \sum^{k}_{i=1} Z^{2}_{i} \sim \chi^{2}(k), \\
f(k) = \frac{1}{2^{k/2} \Gamma (k/2)} x^{k/2-1} e^{-x/2}, \\
F(k) = \frac{1}{\Gamma (k/2)} \gamma (\frac{k}{2},\frac{x}{2})
Explanation
Latex code for the Chi-Squared Distribution. Chi-Squared distribution describes the situation that random variable Q is the sum of k squared value of independent standard normal random variables Z_{1} to Z_{k}.
- Chi-Squared distribution definition: $$Q = \sum^{k}_{i=1} Z^{2}_{i} \sim \chi^{2}(k)$$
- PDF of Chi-Squared distribution: $$\frac{1}{2^{k/2} \Gamma (k/2)} x^{k/2-1} e^{-x/2} $$
- CDF of Chi-Squared distribution: $$\frac{1}{\Gamma (k/2)} \gamma (\frac{k}{2},\frac{x}{2}) $$
- Mean value of Chi-Squared distribution: $$k$$
- Variance value of Chi-Squared distribution: $$2k$$
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