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Mathematics

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Continuous Probability Distributions

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Gamma Distribution

Probability Density Function:

f(x;k,\theta) = \frac{x^{k-1}e^{-x/\theta}}{\theta^k \Gamma(k)}

for x > 0, 0 otherwise. Properties: Mean =

k\theta

, Variance =

k\theta^2

, where Γ(k) is the Gamma function.

Beta Distribution

Probability Density Function:

f(x;\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}

for

0 \leq x \leq 1

, where

B(\alpha,\beta)

is the Beta function. Properties: Bounded by 0 and 1, shape parameters are α and β.

Log-Normal Distribution

Probability Density Function:

f(x;\mu,\sigma) = \frac{1}{x\sigma\sqrt{2\pi}}e^{-\frac{(\ln x-\mu)^2}{2\sigma^2}}

for x > 0. Properties: Logarithm of random variable is normally distributed, Skewed to the right, Multiplicative in nature.

Exponential Distribution

Probability Density Function:

f(x;\lambda) = \lambda e^{-\lambda x}

for x > 0, 0 otherwise. Properties: Memoryless, Mean and Standard Deviation are both equal to

1/\lambda

F-Distribution

Probability Density Function:

f(x;d_1,d_2) = \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1 x + d_2)^{d_1+d_2}}}}{x B(\frac{d_1}{2}, \frac{d_2}{2})}

for x > 0. Properties: Ratio of two scaled chi-squared distributed variables, non-negative, asymmetric and skewed to the right.

Student's t-Distribution

Probability Density Function:

f(x;\nu) = \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})}\left(1+\frac{x^2}{\nu}\right)^{-\frac{\nu+1}{2}}

for all x, where ν is the degrees of freedom. Properties: Symmetrical around zero, heavier tails than normal distribution, approaches the normal distribution as ν increases.

Chi-Squared Distribution

Probability Density Function:

f(x;k) = \frac{x^{\frac{k}{2}-1}e^{-\frac{x}{2}}}{2^{\frac{k}{2}}\Gamma(\frac{k}{2})}

for x > 0, 0 otherwise. Properties: Special case of the Gamma distribution with

\theta = 2

, non-negative, heavily skewed when k is small.

Normal Distribution

Probability Density Function:

f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

Properties: Symmetrical, Bell-Curved, Mean (μ), Median and Mode are all equal.

Weibull Distribution

Probability Density Function:

f(x;\lambda, k) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^k}

for x > 0, 0 otherwise. Properties: Used in survival analysis and reliability engineering, k is the shape parameter, λ is the scale parameter.

Uniform Distribution

Probability Density Function:

f(x) = \frac{1}{b-a}

for

a \leq x \leq b

, 0 otherwise. Properties: All intervals of the same length are equally probable, Mean =

(a+b)/2

, Variance =

(b-a)^2/12

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