Variance + Poisson Distribution

Class: STAT 131 Subject: probability Date: 2025-02-11 Teacher: **Prof. Marcela

Variance

• the variance of a random variable $X$ is $Var(X)=E(X-E(X){)}^2$
• square root of variance is called Standard Deviation(SD) $SD(X)=\sqrt{Var(X)}$

Theorem

• For any random variable $X$, $Var(X)=E(X^2)-(E(X){)}^2$

Properties

• $Var(X+c)=Var(X)$ for any constant c
• $Var(cX)=c^{2}Var(X)$ for any constant c. Variance isn’t linear
• If $X$ and $Y$ are independent, then $Var(X+Y)=Var(X)+Var(Y)$
• $Var(X)\ge 0$, with equality if and only if $P(X=a)=1$ for some constant $a$.

Geometric Distribution

• Let $X$ ~ $Geom(p)$. $E(X)=\frac{q}{p}$ $Var(X)=\frac{q}{p^2}$

Negative Binomial Distribution

• must be independent and identically distributed(iid)
• Let $Y$ ~ $NBin(p)$. $E(Y)=\frac{rq}{p}$ $Var(Y)=\frac{rq}{p^2}$

Binomial Distribution

Let $X$ ~ $Bin(p)$. $E(X)=np$ $Var(X)=np(1-p)$

Poisson Distribution

• a random variable $X$ has the Poisson distribution with parameter $\lambda$, where $\lambda >0$, if the PMF of $X$ is: $P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$, for $k=0,1,2,...$ Written as $X$ ~ $Pois(\lambda )$

• The Poisson distribution is often used in situations where we are counting the number of success in a particular region or interval of time, and there are a large number of trials, each with a small probability of success. Some examples of r.v.s that could follow a distribution that is approx Poisson:

  • Number of emails your receive in an hour.
  • Number of chips in a chocolate chip cookie.
  • Number of earthquakes in a year in some region of the world.

• The parameter can be interpreted as the rate of occurrence of these rare events.

• For example $\lambda =20$ emails per hour, $\lambda =10$ chips per cookie, $\lambda =2$ earthquakes per year

Variance for Poisson Distribution

• Let $X$ ~ $Pois(p)$. $E(X)=\lambda$ $Var(X)=\lambda$