> anishgoyal █


Notes › EENG 3421: Probability and Statistics for Engineers Lecture 13

Variance and Standard Deviation

calendar_today   article 167 words   access_time 2 min   replay Modified

Table of Contents

Variance

Definition 13.1 (Variance).

The variance of random variable $X$ is

$$ \mathrm{Var}[X] = E\left[(X-\mu_X)^2\right] $$
Definition 13.2 (Standard Deviation).

The standard deviation of random variable $X$ is

$$ \sigma_X = \sqrt{\mathrm{Var}[X]} $$
Definition 13.3 (Moments).

For random variable $X$

  • The $n$th moment is
$$ E[X^n] $$
  • The $n$th central moment is
$$ E[(X-\mu_X)^n] $$
Theorem 13.4 (Variance Expansion).
$$ \mathrm{Var}[X] = E[X^2]-\mu_X^2 = E[X^2]-(E[X])^2 $$
Theorem 13.5 (Variance Under Affine Transformations).

For constants $a,b$

$$ \mathrm{Var}[aX+b] = a^2\mathrm{Var}[X] $$
Theorem 13.6 (Variances of Common Families).
  • If $X$ is Bernoulli $(p)$
$$ \mathrm{Var}[X] = p(1-p) $$
  • If $X$ is Geometric $(p)$
$$ \mathrm{Var}[X] = \frac{1-p}{p^2} $$
  • If $X$ is Binomial $(n,p)$
$$ \mathrm{Var}[X] = np(1-p) $$
  • If $X$ is Pascal $(k,p)$
$$ \mathrm{Var}[X] = \frac{k(1-p)}{p^2} $$
  • If $X$ is Poisson $(\alpha)$
$$ \mathrm{Var}[X] = \alpha $$
  • If $X$ is Discrete Uniform $(k,\ell)$
$$ \mathrm{Var}[X] = \frac{(\ell-k)(\ell-k+2)}{12} $$

References

Sources

Graph

mindmap ↗