2. Random Variables

PMF???

Probability Density function

the integral sums to 1 over some range, typically from negative infinity to infinity

Cumulative Distribution Function

$F(x)=P(X\le x)$ $F(\infty )=1$

Discrete vs Continuous

if discrete, then integral is equal to summation from $-\infty$ to x.

Variance

standard deviation

$\sigma =+\sqrt{Var(x)}$

finding a var given random var X and a discrete formula

Could be a geometric series, then the series is equal to - $\frac{1}{1-x}$

Then replace in formula for answer

distributions

discrete

uniform

Bernoulli

Binary probability - success / fail - p / q

binomial (n-bernoulli) (n, p)

probability of x successes in n bernoulli trials

pmf - where the successes happen = $f(x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x{q}^{n-x}$

• trying to find probability of $x$ success
• find number of successes

geometric (p)

model (number of trials) until first success

negative geometric (binomial) distribution (x, r)

model until $rth$ success happens

probability of 5th success in 10 trials

continuous

uniform

$f(x)=\frac{1}{b-a}$

exponential

• e.g. waiting time, time til failure of a machine $E(x)=\frac{1}{lamda}$

normal

exam tips

everything before descriptive statistics is on exam 1

• i.e. covariance is on the exam