In vincenzocoia/distionary: Create and Evaluate Univariate Probability Distributions

d1 <- dst_weibull(1,1) 

d2 <- dst_weibull(1,2)

#d3 <- dst_weibull(1, 2)

shape = 2
scale = 1

sk <- skewness(d2)
var <- variance(d2)
mu <- mean(d2)

$$ \gamma_{2}=\frac{-6 \Gamma_{1}^{4}+12 \Gamma_{1}^{2} \Gamma_{2}-3 \Gamma_{2}^{2}-4 \Gamma_{1} \Gamma_{3}+\Gamma_{4}}{\left[\Gamma_{2}-\Gamma_{1}^{2}\right]^{2}} $$

g1 <- gamma(1 + 1 / shape)
g2 <- gamma(1 + 2 / shape)
g3 <- gamma(1 + 3 / shape)
g4 <- gamma(1 + 4 / shape)
((-6 * g1^4 + 12 * g1^2 * g2 - 3 * g2^2 - 4 * g1 * g3 + g4) /
(g1 - g2^2)^2)


kurtosis_exc.dst(d2)

$$ \gamma_{2}=\frac{\lambda^{4} \Gamma\left(1+\frac{4}{k}\right)-4 \gamma_{1} \sigma^{3} \mu-6 \mu^{2} \sigma^{2}-\mu^{4}}{\sigma^{4}}-3 $$

# d <- dst_weibull(1,1)
# sk <- skewness(d)
# var <- variance(d)
# mu <- mean(d)
# ((gamma(1+4/1) - 4*sk*(var^3)*mu - 6*(mu^2)*(var^2) - mu^4)/(var^4) - 3)


g1 <- gamma(1 + 1 / shape)
g2 <- gamma(1 + 2 / shape)
g3 <- gamma(1 + 3 / shape)
g4 <- gamma(1 + 4 / shape)
mu <- scale * g1
var <- scale^2 * (g2 - g1^2)
sigma <- sqrt(var)


sk <- (g3 * scale^3 - 3 * mu * var - mu^3) / sigma^3
((gamma(1+4/1) - 4*sk*(var^3)*mu - 6*(mu^2)*(var^2) - mu^4)/(var^4) - 3)

kurtosis_exc.dst(d2)

$$ \gamma_{2} = \frac{\Gamma\left(1+\frac{4}{\tau}\right) \lambda^{4}-4 \mu \Gamma\left(1+\frac{3}{\tau}\right) \lambda^{3}+6 \mu^{2} \Gamma\left(1+\frac{2}{\tau}\right) \lambda^{2}-3 \mu^{4}}{\sigma^{4}} -3 $$

var <- variance(d2)
mu <- mean(d2)

((gamma(1 +  4/shape)*(scale^4) - 4*mu*gamma(1 + 3/shape)*(scale^3) + 6*(mu^2)*gamma(1+2/shape)*(scale^2) - 3*(mu^4))/(var^4) - 3)

vincenzocoia/distionary documentation built on March 5, 2024, 3:13 a.m.