Discrete random variables

statistics video

Probability distribution for X:

x	0	1	2
p(x)	.16	.48	.36

X (uppercase) is the random variable. x (lowercase) is the value that X can take on. p(x) is the probability of hitting that number.

Expected Value

Expected value is the therotetical mean (based on the distribution).

$\mu =E(X)={\sum }_{allx}xp(x)$

E = expected value, which is the same thing as “mean”.

Population Variance of discrete random variables

Variance of X, sometimes written as ${\sigma }^2$:

$E[(X-\mu {)}^2]={\sum }_{allx}(x-\mu {)}^{2}p(x)$

These relate:

$E[(X-\mu {)}^2]=E(X^2)-[E(X){]}^2$

with symbols:

${\sigma }^2=E(X^2)-{\mu }^2$

Examples

Expectation of the probability table:

$E(X)=\mu =0\cdot 0.16+1\cdot .48+2\cdot .36=1.2$ $E(X^2)=0^2\cdot 0.16+1^2\cdot .48+2^2\cdot .36=1.92$

$$\begin{array}{r}E((X-\mu {)}^2]=(0-1.2{)}^2\cdot .16+\\ \\ (1-1.2{)}^2\cdot .48+\\ \\ (2-1.2{)}^2\cdot .36\\ \\ =0.48={\sigma }^2\end{array}$$

stddev = $\sigma$ = $\sqrt{0.48}$

Mean, Variance and Standard Deviation for Binomial Random Values

n = count of trials p = probability of success q = probability of failure (aka $1-p$)

Mean: $\mu =np$ Variance: ${\sigma }^2=npq$ Standard deviation: $\sigma =\sqrt{npq}$