discrete probability

measuring spread

Consider a random variable. We would like to measure how spread out the values are from the expected value, aka the mean, $E(X)$.

x	1	2	3
Pr(X=x)	1/3	1/3	1/3

In this case, $E(X)=2$. we can find the difference, and square that so the distance from the expected value is positive.

x	1	2	3
Pr(X=x)	1/3	1/3	1/3
x-E(X)	-1	0	1
(x-E(X))^2	1	0	1

Therefore, the variance of $X$, $\text{Var}(X)=E((x-E(X){)}^2)$. We can evaluate it to be $1/3\cdot 1+1/3\cdot 0+1/3\cdot 1=\frac{2}{3}$

Good news, we can find an easier way to find the variance. Hooray.

$\text{Var}(X)=E((x-E(X){)}^2)$
$=E(X^2-2X\cdot E(X)+E(X{)}^2)$
$=E(X^2)-2E(E(X{)}^2)+E(X{)}^2$
$\text{Var}(X)=E(X^2)-E(X{)}^2$

id: 1752206438673
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$\text{Var}(X)$ = ==$E(X^2)-E(X)^2$==

binomial distribution and binomial experiment

A binomial experiment (aka Bernoulli sequence) is a set of trials where:

• Each trial results in a binary outcome: A or A’
• All trials have the same probability for the outcomes.

Input $n$ is the number of trials. Input $p$ is probability of success, $\text{Pr(A)}$.

The statement below says “the random variable $X$ is distributed as a binomial distribution with 3 trials and 0.1 probability of success”.
$X\sim \text{Bi}(n=3,p=0.1)$

The probability of achieving $x$ successes in $n$ trials of a binomial experiment is:

$\text{Pr}(X=x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x}$

The probability of $X$ being within the range $[a,b]$ is calculated with $\text{binomCdf}(n,p,a,b)$

For example, $\text{Pr}(X\ge 30)=\text{binomCdf}(n,p,30,\infty )$

exercise - binomial experiment
exercise - discrete random variables