Deterministic: No randomness involved - concept of probability N/A

Sample Space: Set of possible outcomes of a random experiment; denoted $S$

• Discrete: Has finite or countably infinite (ℤ) number of events
• Events: Subsets of sample space; simple if size = 1, compound if size > 1

Uniform Probability Model: Each event in $S$ is equally likely

• $P(A)=\frac{\text{number of outcomes in }A}{\text{number of elements in }S}$
• AND = multiplication, OR = addition

Permutations: Arrangements of sequences $~_nP_k = n^{(k)}$

• Assume we have a list of $n$ symbols
• $n! =$ # of arrangements using each symbol exactly once
• $n^{(k)}=\frac{n!}{(n-k)!} =$ # of arrangements of length k, using each symbol at most once
• $n^k=$ # of arrangements of length k, using each symbol any amount of times

Sterling's Approximation: Approximates $n!≈(\frac{n}{e})^n\sqrt{2πn}$

Combinations: Arrangements, order is arbitrary (i.e. "sort then remove dupes")

• ${n \choose k} = \frac{n^{(k)}}{k!}=\frac{n!}{k!(n-k)!}=$ # of subsets of size k that can be selected from a set with size n
  • "How many ways can we choose k elements from n elements without replacement?"

Random Variables: Function from sample space → ℝ (e.g. $X=\text{the number of heads when a coin is flipped thrice}$)

• Previously, we use $A$ to represent the set of possible values for $X$ (e.g. $A={0, 1, 2, 3}$)
• Discrete: Countable amount of outcomes; Continuous: Non-countable amount of outcomes
• Probability Function: $f(x)=P(X=x)=F(x) - F(x-1)$
• Cumulative Distribution Function: $F(x)=P(X≤x)=\sum_{y≤x}f(y)$
  • Non-decreasing; F(x) ∈ [0, 1]; limit as $x→-∞$ is 0, limit to $∞$ is 1
• For graphs of probability functions, the probability is given by the area under the curve

1 & 2

3

3.2

3.3

Binomial Distribution: Two outcomes: "success" or "failure".

$f(x)=P(X=x)={n\choose x}p^x(1-p)^{n-x}\text{ for }x=0, 1, 2, \ldots, n$

• $p$ probability of success, $n$ experiments (a.k.a. Bernoulli trials)
• Each event is independent (has replacement) - the probability is constant
• Wolfram: **X ~ B(n, p)**, X is number of successes

Hypergeometric Distribution: Binomial without replacement

$f(x)=P(X=x)={r \choose x}{N-r \choose n-x}\div{N \choose n}$

• $N$ objects, $r$ "success" objects, $N-r$ "failure" objects
• Pick $n$ objects without replacement - this indicates that the probability is not constant
• Wolfram: **X ~ HypergeometricDistribution(n, r, N)**, X is number of successes

Negative Binomial Distribution: Binomial, except we keep doing the experiment until we get $k$ successes

$f(x)=P(X=x)={x+k-1\choose x}p^k(1-p)^x\text{ for }x=0, 1, 2, \ldots,$

• $p$ probability of success, $k$ desired number of successes
• Wolfram: **X ~ NB(k, p)**, X is number of failures before $k$th success

Geometric Distribution: Negative Binomial with $k=1$ success

$f(x)=P(X=x)=p(1-p)^x\text{ for }x=0, 1, 2, \ldots,$

• Wolfram: X ~ NB(1, p), X is number of failures before a success

Poisson Distribution from Binomial: Binomial as $n→∞, p→0$

$f(x)=μ^xe^{-p}\div x! \text{ for } x =0, 1, 2, \ldots$

• Increase n, decrease p, fix $μ=np$
• Approximates Binomial with large $n$ and small $p$
• Wolfram: X ~ Poisson(μ), X is number of events

Poisson Distribution from Poisson Process: Events that occur at points in time or space

$f_t(x)=\frac{μ^xe^{-μ}}{x!}\text{ for }x=0,1,2,\ldots$

• $μ=λt$, $λ$ rate of occurrence, $t$ length of time interval