Student t-Distribution
Tags: #Math #StatisticsEquation
$$f(t) = \frac{\Gamma(\frac{v + 1}{2})}{\sqrt{v\pi}\Gamma(v/2)} (1+\frac{t^{2}}{v})^{-(v+1)/2}, \\ F(t) = \frac{1}{2} + x \Gamma(\frac{v+1}{2}) \times \frac{2^{F_{1}}(1/2,\frac{v+1}{2};3/2;-\frac{x^{2}}{v})}{\sqrt{\pi v} \Gamma(\frac{v}{2})}, \\ F(t) = 1 - \frac{1}{2} I_{x(t)} (\frac{v}{2}, \frac{1}{2}), x(t)=\frac{v}{t^{2} + v}$$Latex Code
f(t) = \frac{\Gamma(\frac{v + 1}{2})}{\sqrt{v\pi}\Gamma(v/2)} (1+\frac{t^{2}}{v})^{-(v+1)/2}, \\
F(t) = \frac{1}{2} + x \Gamma(\frac{v+1}{2}) \times \frac{2^{F_{1}}(1/2,\frac{v+1}{2};3/2;-\frac{x^{2}}{v})}{\sqrt{\pi v} \Gamma(\frac{v}{2})}, \\
F(t) = 1 - \frac{1}{2} I_{x(t)} (\frac{v}{2}, \frac{1}{2}), x(t)=\frac{v}{t^{2} + v}
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Introduction
Equation
$$f(t) = \frac{\Gamma(\frac{v + 1}{2})}{\sqrt{v\pi}\Gamma(v/2)} (1+\frac{t^{2}}{v})^{-(v+1)/2} $$ $$F(t) = \frac{1}{2} + x \Gamma(\frac{v+1}{2}) \times \frac{2^{F_{1}}(1/2,\frac{v+1}{2};3/2;-\frac{x^{2}}{v})}{\sqrt{\pi v} \Gamma(\frac{v}{2}) } $$ $$F(t) = 1 - \frac{1}{2} I_{x(t)} (\frac{v}{2}, \frac{1}{2}), x(t)=\frac{v}{t^{2} + v}$$
Latex Code
f(t) = \frac{\Gamma(\frac{v + 1}{2})}{\sqrt{v\pi}\Gamma(v/2)} (1+\frac{t^{2}}{v})^{-(v+1)/2}, \\
F(t) = \frac{1}{2} + x \Gamma(\frac{v+1}{2}) \times \frac{2^{F_{1}}(1/2,\frac{v+1}{2};3/2;-\frac{x^{2}}{v})}{\sqrt{\pi v} \Gamma(\frac{v}{2})}, \\
F(t) = 1 - \frac{1}{2} I_{x(t)} (\frac{v}{2}, \frac{1}{2}), x(t)=\frac{v}{t^{2} + v}
Explanation
Latex code for the Student t-Distribution. The Student t-Distribution t_{v} is a continuous distribution that generalize the standard normal distribution. But the t-Distribution t_{v} has heavier tails.
- PDF of Student t-Distribution: $$f(t) = \frac{\Gamma(\frac{v + 1}{2})}{\sqrt{v\pi}\Gamma(v/2)} (1+\frac{t^{2}}{v})^{-(v+1)/2}$$
- CDF of Student t-Distribution: $$F(t) = \frac{1}{2} + x \Gamma(\frac{v+1}{2}) \times \frac{2^{F_{1}}(1/2,\frac{v+1}{2};3/2;-\frac{x^{2}}{v})}{\sqrt{\pi v} \Gamma(\frac{v}{2})}$$
- Hypergeometric function : $$2^{F_{1}}(a,b;c;z)$$
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