Stochastic Modeling

Basic Definitions¶

Random Events¶

Occurrence: The event $A$ occurs if the outcome of the experiment belongs to $A$.

Special Events¶

• $A \cap B$: $A$ and $B$ (intersection)

• $A \cup B$: $A$ or $B$ (non-exclusive “or”)

• $B \setminus A$: $B$ without $A$ (set difference)

• $A^c = \Omega \setminus A$: the complementary event “not $A$”

• $\bigcap_{i=1}^\infty A_i$: all $A_i$ occur

• $\bigcup_{i=1}^\infty A_i$: at least one $A_i$ occurs

Probability Measure¶

Conditional Probabilities¶

Independence¶

Informally, two events are independent if the occurrence of one does not influence the other.

Properties of Independence¶

1. $E$ and $F$ are independent if $P(E|F) = P(E)$ or $P(F|E) = P(F)$.

2. If $E$ and $F$ are independent, then their complements ($E^c, F^c$) are also independent.

Random Variables¶

A random variable (RV) $X$ is a function that maps the sample space of a random experiment to the real numbers: $X: \Omega \to \mathbb{R}$.

Classification of RVs¶

Moments and Functions¶

The following quantities describe the behavior of a random variable $X$:

• Distribution Function (CDF): $F_X(t) = P(X \le t)$

• Expectation (Mean): $E[X] = \int x f_X(x) dx$ (or sum for discrete)

• Variance: $Var(X) = E[X^2] - (E[X])^2$

• Moment Generating Function (MGF): $\phi_X(t) = E[e^{tX}]$

Discrete Probability Distributions¶

Bernoulli Distribution¶

Models a single trial with success probability $p$.

• Notation: $X \sim \text{Ber}(p)$

• $E[X] = p$

• $Var(X) = p(1-p)$

Binomial Distribution¶

Models the number of successes in $n$ independent Bernoulli trials.

• Notation: $X \sim \text{Bin}(n, p)$

• PMF: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$

• $E[X] = np$

• $Var(X) = np(1-p)$