Random Number Distributions¶

Uniform Distribution¶

rand()¶

Generates a random number between \(0\) and \(1\) from the uniform distribution \(U(0, 1)\).

>>> rand()
= 0.1629098753910512

Gaussian Distribution¶

randn()¶

Generates a random number from the gaussian distribution \(\mathcal{N}(0, 1)\).

The probability distribution for Gaussian random variates is,

\[p(x) dx = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left( -\frac{x^2}{2\sigma^2} \right) dx\]

>>> randn()
= 0.1339186081186759

gauss_pdf(x, sigma=1)¶

Computes the probability density \(p(x)\) at \(x\) for a Gaussian distribution with standard deviation sigma, using the formula given above.

>>> gauss_pdf(2)
= 0.05399096651318806
>>> gauss_pdf(2, 1)
= 0.05399096651318806

gauss_P(x, sigma=1)¶

Computes the cumulative density \(P(x)\) at \(x\)

\[P(x) = \int_{-\infty}^{x} p(t) dt\]

where \(p(t)\) is the gaussian distribution with standard deviation sigma. Aliases: norm

>>> gauss_P(2)
= 0.9772498680518208
>>> norm(2, 1)
= 0.9772498680518208

gauss_Pinv(x, sigma=1)¶

Computes the inverse of the cumulative density \(P^{-1}(x)\) at \(x\) for the gaussian distribution with standard deviation sigma. Aliases: norminv

>>> gauss_Pinv(0.9772498680518208)
= 2
>>> norminv(2, 1)
= 2

gauss_Q(x, sigma=1)¶

Computes the cumulative density \(P(x)\) at \(x\)

\[Q(x) = \int_{x}^{\infty} p(t) dt\]

where \(p(t)\) is the gaussian distribution with standard deviation sigma. Aliases: cnorm

>>> gauss_Q(2)
= 0.02275013194817921
>>> cnorm(2, 1)
= 0.02275013194817921

gauss_Qinv(x, sigma=1)¶

Computes the inverse of the cumulative density \(Q^{-1}(x)\) at \(x\) for the gaussian distribution with standard deviation sigma. Aliases: cnorminv

>>> gauss_Qinv(0.02275013194817921)
= 2
>>> cnorminv(2, 1)
= 2

Exponential Distribution¶

exponential(mu)¶

Generates a random number from the exponential distribution with mean mu.

The probability distribution for exponential random variates is,

\[p(x) dx = \frac{1}{\mu} \exp\left( -\frac{x}{\mu} \right) dx\]

for \(x \ge 0\)

>>> exponential(1)
= 8.261578216370394
>>> exponential(1)
= 0.177823538531874

exponential_pdf(x, mu)¶

Computes the probability density \(p(x)\) at \(x\) for an exponential distribution with mean mu, using the formula given above.

>>> exponential_pdf(2, 1)
= 0.1353352832366127I

exponential_P(x, mu)¶

Computes the cumulative density \(P(x)\) at \(x\)

\[P(x) = \int_{-\infty}^{x} p(t) dt\]

where \(p(t)\) is the exponential distribution with mean mu. Aliases: exp_P

>>> exponential_P(2, 1)
= 0.8646647167633873
>>> exp_P(2, 1)
= 0.8646647167633873

exponential_Pinv(x, mu)¶

Computes the inverse of the cumulative density \(P^{-1}(x)\) at \(x\) for the exponential distribution with mean mu. Aliases: exp_Pinv

>>> exponential_Pinv(0.8646647167633873, 1)
= 2
>>> exp_Pinv(0.8646647167633873, 1)
= 2

exponential_Q(x, mu)¶

Computes the cumulative density \(P(x)\) at \(x\)

\[Q(x) = \int_{x}^{\infty} p(t) dt\]

where \(p(t)\) is the exponential distribution with mean mu. Aliases: exp_Q

>>> exponential_Q(2, 1)
= 0.1353352832366127
>>> exp_Q(2, 1)
= 0.1353352832366127

exponential_Qinv(x, mu)¶

Computes the inverse of the cumulative density \(Q^{-1}(x)\) at \(x\) for the exponential distribution with mean mu. Aliases: exp_Pinv

>>> exponential_Qinv(0.1353352832366127, 1)
= 2
>>> exp_Qinv(0.1353352832366127, 1)
= 2

Chi-Squared Distribution¶

chisq(nu)¶

Generates a random number from the chi-squared distribution with nu degrees of freedom.

The probability distribution for chi-squared random variates is,

\[p(x) dx = \frac{1}{2\Gamma(\nu/2)} (x/2)^{\nu/2-1} \exp\left( -\frac{x}{2} \right) dx\]

for \(x \ge 0\)

>>> chisq(1)
= 1.915412399185662
>>> chisq(1)
= 3.098215236623985

chisq_pdf(x, nu)¶

Computes the probability density \(p(x)\) at \(x\) for an chi-squared distribution with nu degrees of freedom, using the formula given above.

>>> chisq_pdf(2, 1)
= 0.1037768743551486

chisq_P(x, nu)¶

Computes the cumulative density \(P(x)\) at \(x\)

\[P(x) = \int_{-\infty}^{x} p(t) dt\]

where \(p(t)\) is the chi-squared distribution with nu degrees of freedom.

>>> chisq_P(2, 1)
= 0.842700792949715
>>> chisq_P(2, 2)
= 0.6321205588285578

chisq_Pinv(x, nu)¶

Computes the inverse of the cumulative density \(P^{-1}(x)\) at \(x\) for the chi-squared distribution with nu degrees of freedom.

>>> chisq_Pinv(0.842700792949715, 1)
= 2
>>> chisq_Pinv(0.6321205588285578, 2)
= 2

chisq_Q(x, nu)¶

Computes the cumulative density \(P(x)\) at \(x\)

\[Q(x) = \int_{x}^{\infty} p(t) dt\]

where \(p(t)\) is the chi-squared distribution with nu degrees of freedom.

>>> chisq_Q(2, 1)
= 0.157299207050285
>>> chisq_Q(2, 2)
= 0.3678794411714423

chisq_Qinv(x, nu)¶

Computes the inverse of the cumulative density \(Q^{-1}(x)\) at \(x\) for the chi-squared distribution with nu degrees of freedom.

>>> chisq_Qinv(0.157299207050285, 1)
= 2
>>> chisq_Qinv(0.3678794411714423, 2)
= 2