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Distributions – probability formulas

Distributions

Normal distribution:

The normal distribution (also called Gaussian distribution) is the most used statistical distribution because of the many physical, biological, and social processes that it can model.

The probability density function (PDF) is:

$$ f_X(x)= \frac {1}{\sigma \sqrt {2\pi}} e^{-\frac {(x-\mu)^2}{2 \sigma^2} } $$

Where,

  • X is the random variable.
  • μ is the mean value.
  • σ is the standard deviation (std) value.

Binomial distribution:

The binomial distribution is used to represent the number of events that occurs within n independent trials. Possible values are integers from zero to n.

The probability mass function (PMF) is:

$$ f(x)= \binom n x p^x(1-p)^{n-x}, x=0,1,2,\cdots,n $$

Where,

  • \( \binom n x \) equals \( \frac {n!}{x!(n-x)!} \)
  • mean = np
  • variance = np(1 – p)

Poisson distribution:

The Poisson distribution is a discrete distribution that models the number of events based on a constant rate of occurrence. The Poisson distribution can be used as an approximation to the binomial when the number of independent trials is large and the probability of success is small.

The probability mass function (PMF) is:

$$ f(x)= \frac {e^{-\lambda}\lambda^x}{x!},x=0,1,2,\cdots, \lambda \gt 0 $$

Where,

  • mean = λ
  • variance = λ

Exponential distribution:

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution.

Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

$$ F(x) = \lambda e^{- \lambda x} $$

Where,

  • mean = \( \frac {1}{\lambda} \)
  • variance =\( \frac {1}{\lambda^2} \)

Beta distribution:

The beta distribution is often used to represent processes with natural lower and upper limits.

The probability density function (PDF) is:

$$ f(x)= \frac { \Gamma(\alpha+ \beta)}{\Gamma \alpha \Gamma \beta}\frac{ (x-a)^{\alpha-1}(b-x)^{\beta-1} }{(b-a)^{\alpha+ \beta-1}}, a \le x \le b, a \lt b,\alpha \gt 0,\beta \gt 0 $$

Where,

  • mean= \( \frac {\alpha}{\alpha+ \beta} \)
  • variance= \( \frac {\alpha \beta}{(\alpha+ \beta+ 1)(\alpha+ \beta)^2} \)

Chi-square distribution:

If X has a standard normal distribution, X2 has a chi-square distribution with one degree of freedom, allowing it to be a commonly used sampling distribution.

The sum of n independent X2 variables (where X has a standard normal distribution) has a chi-square distribution with n degrees of freedom. The shape of the chi-square distribution depends on the number of degrees of freedom.

The probability density function (PDF) is:

$$ f(x)= \frac {x^{\frac {(\nu-2)}{2}}e^{-\frac x2}}{2^{\frac {\nu}{2}} \Gamma \left(\frac {\nu}{2}\right)}, x \gt 0, \nu \gt 0 $$

Where,

  • mean = v
  • variance = 2v
  • ν-degrees of freedom
  • Γ-gamma function
  • e-base of the natural logarithm

F-distribution:

The F-distribution is also known as the variance-ratio distribution and has two types of degrees of freedom: numerator degrees of freedom and denominator degrees of freedom. It is the distribution of the ratio of two independent random variables with chi-square distributions, each divided by its degrees of freedom.

The probability density function (PDF) is:

$$ f(x)=\frac {\Gamma \left( \frac {v+u}{2} \right)}{\Gamma \left( \frac u2\right) \Gamma \left( \frac v2\right)} \left( \frac uv\right)^{\frac u2} \frac {x^{\frac {u-2}{2}}}{1+ \left( \frac uv x\right)^{\frac {u+v}{2}}},x \gt 0, u \gt 0, v \gt 0 $$

Where,

  • mean =\( \frac {v}{v-2}, v \gt 2 \)
  • variance =\( \frac {2v^2(u+v-2)}{u(v-2)^2(v-4)}, v \gt 4 \)
  • Γ – gamma function
  • u – numerator degrees of freedom
  • v – denominator degrees of freedom

Gamma distribution:

The gamma distribution is often used to model positively skewed data.

The probability density function (PDF) is:

$$ f(x)= \frac {(x-\theta)^{a-1} e^{-\frac {(x-\theta)}{b}}}{\Gamma(a) b^a}, x \gt \theta, a \gt 0, b\gt 0 $$

Where,

  • mean =\( ab+\theta \)
  • variance =\( ab^2 \)
  • a – shape parameter
  • b – scale parameter
  • θ – threshold parameter
  • Γ – gamma function
  • e – base of the natural logarithm

Uniform distribution:

The uniform distribution characterizes data over an interval uniformly, with a as the smallest value and b as the largest value.

The probability density function (PDF) is:

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

Where,

  • mean =\( \frac {a+b}{2} \)
  • variance =\( \frac {(b-a)^2}{12} \)
  • a – lower endpoint
  • b – upper endpoint

Laplace distribution:

The Laplace distribution is used when the distribution is more peaked than a normal distribution.

The probability density function (PDF) is:

$$ f(x)=\frac {1}{2b} e^{-\frac {|x-a|}{b}}, b \gt 0, -\infty \lt x \lt \infty , -\infty \lt a \lt \infty $$

Where,

  • mean =\( a \)
  • variance =\( 2b^2 \)
  • a – location parameter
  • b – scale parameter

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