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Variance and Standard Deviation in Probability

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Discrete Random Variable with Finite Outcomes

Variance $(\sigma^2)$ is the sum of the product of each outcome's probability and the square of its deviation from the mean. $\sigma = \sqrt{\sigma^2}$

Continuous Random Variable with Probability Density Function (PDF)

\sigma^2 = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) dx

Where

f(x)

is the PDF and

\mu

is the mean.

\sigma = \sqrt{\sigma^2}

Binomial Random Variable (n independent trials, each with success probability p)

\sigma^2 = np(1-p)

where

n

is the number of trials and

p

is the success probability.

\sigma = \sqrt{np(1-p)}

Poisson Random Variable (events in fixed interval with mean rate $\lambda$ )

Variance $(\sigma^2)$ is equal to the mean rate $\lambda$ . $\sigma = \sqrt{\lambda}$

Uniform Random Variable (continuous, with range $a$ to $b$ )

\sigma^2 = \frac{(b - a)^2}{12}

where

a

and

b

are the minimum and maximum values, respectively.

\sigma = \sqrt{\frac{(b - a)^2}{12}}

Exponential Random Variable (time until event with rate $\lambda$ )

Variance $(\sigma^2)$ is given by $\frac{1}{\lambda^2}$ . $\sigma = \frac{1}{\lambda}$

Normal Random Variable (mean $\mu$ and variance $\sigma^2$ )

Variance is given by parameter $\sigma^2$ . Standard Deviation is $\sigma$

Geometric Random Variable (number of trials until first success, with success probability p)

\sigma^2 = \frac{1-p}{p^2}

Standard Deviation

\sigma = \sqrt{\frac{1-p}{p^2}}

Hypergeometric Random Variable (successes in samples without replacement)

\sigma^2 = \frac{N-K}{N-1} * \frac{K}{N} * \frac{N-n}{N} * n

where

N

is the population size,

K

is the number of success states in the population, and

n

is the sample size.

Negative Binomial Random Variable (number of trials until r-th success)

\sigma^2 = \frac{r(1-p)}{p^2}

where

r

is the number of required successes and

p

is the probability of success on each trial. Standard Deviation

\sigma = \sqrt{\frac{r(1-p)}{p^2}}

Multinomial Random Variable (n independent trials, each with k possible outcomes)

For any two outcomes $i$ and $j$ , $Cov(X_i, X_j) = -np_ip_j$ if $i \neq j$ . For outcome $i$ , $Var(X_i) = np_i(1-p_i)$ . Standard Deviation $\sigma_i = \sqrt{np_i(1-p_i)}$

Chi-Squared Random Variable (sum of squares of k standard normal random variables)

Variance is $2k$ where $k$ is the degrees of freedom. Standard Deviation $\sigma = \sqrt{2k}$

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