x̄ - > Hyperbolic distribution shape=1 skewness=0 scale=1 location=0

The hyperbolic distribution is a continuous probability distribution that has four parameters: shape, skewness, scale, and location. The specific values you provided for each parameter (shape=1, skewness=0, scale=1, location=0) define a particular instance of the hyperbolic distribution.

The hyperbolic distribution with these parameters has a probability density function (PDF) that can be written as:

f(x; shape=1, skewness=0, scale=1, location=0) = (2/π) * (1/sqrt(shape^2 + 1)) * exp(-sqrt(shape^2 + 1) * abs(x-location) + skewness*(x-location))

where x is a random variable, and abs(x-location) denotes the absolute value of the difference between x and the location parameter.

With shape=1, this distribution has heavier tails than the normal distribution, meaning that extreme values are more likely to occur. Skewness=0 means that the distribution is symmetric around its mean, which is equal to the location parameter (in this case, 0). Scale=1 sets the spread of the distribution, and location=0 sets the center of the distribution at 0.

Overall, the hyperbolic distribution with these parameters has some interesting properties and could be useful in modeling certain types of data.

hyperbolic distribution shape=1 skewness=0 scale=1 location=0

hyperbolic distribution | shape | α = 1

skewness parameter | β = 0

scale | δ = 1

location | μ = 0hyperbolic distribution | shape | α = 1

skewness parameter | β = 0

scale | δ = 1

location | μ = 0

mean | 0

standard deviation | sqrt((K_2(1))/(K_1(1)))≈1.64301

variance | (K_2(1))/(K_1(1))≈2.69948

skewness | 0

kurtosis | (3 K_1(1) K_3(1))/K_2(1)^2≈4.85697

e^(-sqrt(x^2 + 1))/(2 K_1(1))

0.854524 | 0.874223 | 0.0741282 | -3.83045 | -0.43346

10th | -1.93997

25th | -0.924464

50th | -3.34835×10^-14

75th | 0.924464

90th | 1.93997