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Mathematics

Probability

Continuous Random Variables

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The height of grown men in a certain city, assuming it's normally distributed.

Normal (Gaussian) distribution PDF: $f(x | \mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$

The time it takes for a randomly selected computer part to fail, following an exponential distribution.

Exponential distribution PDF: $f(x | \lambda) = \lambda e^{-\lambda x}$ for $x \geq 0$

The lifespan of a particular species of fruit flies, which is known to follow a uniform distribution over the interval from 30 to 50 days.

Uniform distribution PDF: $f(x | a, b) = \frac{1}{b-a}$ for $a \leq x \leq b$

The amount of milk a cow produces in a day, which follows a skewed distribution due to varying genetic and environmental factors.

Gamma distribution PDF: $f(x | k, \theta) = \frac{x^{k-1}e^{-\frac{x}{\theta}}}{\theta^k \Gamma(k)}$ for $x > 0$

The time in hours a certain brand of light bulb operates before it burns out.

Weibull distribution PDF: $f(x | k, \lambda) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-\left(\frac{x}{\lambda}\right)^k}$ for $x \geq 0$

Analyzing the daily river flow rates, which exhibit significant variability but can be modeled using a log-normal distribution.

Log-normal distribution PDF: $f(x | \mu, \sigma) = \frac{1}{x\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{\ln x - \mu}{\sigma})^2}$ for $x > 0$

The concentration of a pollutant in a body of water, typically exhibiting high levels of kurtosis.

Pareto distribution PDF: $f(x | k, x_m) = \frac{k x_m^k}{x^{k+1}}$ for $x \geq x_m$

The speed of cars on a freeway, where the speeds are constrained between a minimum and a maximum value, can be approximated by a triangular distribution.

Triangular distribution PDF: Conditional distributions: $\begin{cases} f(x | a, b, c) = 2(x-a)/((b-a)(c-a)) & \text{for } a \leq x < c, \\ f(x | a, b, c) = 2(b-x)/((b-a)(b-c)) & \text{for } c \leq x \leq b, \end{cases}$

A physicist is studying the energy level of a quantum harmonic oscillator, which can be described by the Rayleigh distribution.

Rayleigh distribution PDF: $f(x | \sigma) = \frac{x}{\sigma^2}e^{-\frac{x^2}{2\sigma^2}}$ for $x \geq 0$

The wind speed at a certain location, assuming there are no upper limits and the speeds follow a particular long-tailed distribution.

Gumbel distribution PDF for the maximum (Type I Extreme Value Distribution): $f(x | \mu, \beta) = \frac{1}{\beta} e^{-\frac{x - \mu}{\beta} - e^{-\frac{x - \mu}{\beta}}}$

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