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probability and stochastic processes

Probabilistic Systems Analysis and Applied Probability

1. experiments, models, and probabilities

theorem 1.1 demorgan's law related all three basic operations $(A \cup B)^c = (A^c \cap B^c)$

theorem 1.2 for mutually exclusive events $A_1$ and $A_2$, $P[A_1 \cup A_2] = P[A_1] + P[A_2]$

theorem 1.3 $text{If } A = A_1 \cup A_2 \cup \cdots \cup A_m \text{ and } A_i \cap A_j = \emptyset \text { for all } i \neq j \text{ , then }$

$$ P[A] = \sum_{i=1}^m P[A_i] $$

theorem 1.4 The probability measure $P[.]$ is a function that satisfies the following properties:

• $P[\emptyset] = 0$

• $P[A^c] = 1 - P[A]$

• For any A and B (not necessarily mutually exclusive), $P[A \cup B] = P[A] + P[B] - P[A \cap B]$

• $A \subset B, P[A] \leq P[B]$

Theorem 1.5 The probability of an event $B = {s_1, s_2, \cdots, s_m}$ is the sum of the probabilities of the outcomes contained in the event:

$$ P[B] = \sum_{i=1}^m P[{s_i}] $$

theorem 1.6 For an experiment with sample space $S = {s_1, s_2, \cdots, s_n}$ in which each outcomes $s_i$ is equally likely,

$$P[{s_i}] = \frac{1}{n} \space \space \space 1 ≤ i ≤ n$$

theroem 1.7 A conditional probability measure $P[A|B]$ has the following properties that correspond to the axioms of probability:

Axiom 1: $P[A|B] \geq 0$

Axiom 2: $P[B|B] = 1$

Axiom 3: If $A = A_1 \cup A_2 \cup \cdots \cup A_m$ and $A_i \cap A_j = \emptyset$ for all $i \neq j$, then

$$P[A|B] = P[A_1|B] + P[A_2|B] + \cdots + P[A_m|B]$$

Theorem 1.8 For a partition $B = {B_1, B_2, \cdots, B_m}$ and any event $A$ in the sample space, let $C_i = A \cap B_i$ For $i ≠ j$, the events $C_i$ and $C_j$ are mutually exclusive and $A = C_1 \cup C_2 \cup \cdots$

Theorem 1.9 For any event $A$ and partition ${B_1, B_2, \cdots, B_m}$

$$P[A] = \sum_{i=1}^m P[A \cap B_i$$

Theorem 1.10 Law of total probability

For a partition ${ B_1, B_2, \cdots, B_m }$ with $P[B_i] > 0$ for all $i$,

$$ P[A] = \sum_{i=1}^m P[A|B_i] P[B_i] $$

Theorem 1.11 Bayes' theorem

$$ P[B|A] = \frac{P[A|B] P[B]}{P[A]} $$

Definition 1.1 Outcome An outcome of an experiment is a possible result of the experiment.

Definition 1.2 Sample space The sample space of an experiment is the finest-grain, mutually exclusive, collectively exhaustive set of all possible outcomes of the experiment.

Definition 1.3 Event An event is a subset of the sample space.

Definition 1.4 Axioms of Probability A probability measure $P[.]$ is a function that maps events in the sample spacce to real numbers such that

Axiom 1 For any event $A$, $P[A] \geq 0$

Axiom 2 $P[S] = 1$

Axiom 3 For any countable collection $A_1, A_2, \cdots$ of mutually exclusive events,

$$ P[A_1 \cup A_2 \cup \cdots] = P[A_1] + P[A_2] + \cdots $$

Definition 1.5 Conditional probability The conditional probability of an event $A$ given the occurance of the event B is

$$ P[A|B] = {P[AB] \over P[B]}$$

Conditional probability is defined only when $P[B] > 0$.

Definition 1.6 Two independent events Two events $A$ and $B$ are independent if

$$ P[AB] = P[A]P[B] $$

Definition 1.7 Three Independent Events $A_1, A_2, A_3$ are mutually exclusive and independent if and only if

(a) $A_1$ and $A_2$ are independent

(b) $A_2$ and $A_3$ are independent

(c) $A_1$ and $A_3$ are independent

(d) $P[A_1 \cap A_2 \cap A_3] = P[A_1]P[A_2]P[A_3]$

Definition 1.8 More than Two Independent Events

If $n ≥ 3$ events $A_1, A_2, \cdots, A_n$ are mutually independent if an only if

(a) all collections of $n - 1$ events chosen from $A_1, A_2, \cdots, A_n$ are mutually independent,

(b) $P[A_1 \cap A_2 \cap \cdots \cap A_n] = P[A_1]P[A_2] \cdots P[A_n]$

2. Sequential Experiments

Theorem 2.1 An experiment consists of two subexperiments. If one subexperiment has $k$ outcomes and the other has $n$ outcomes, then the experiment has $kn$ outcomes.

Theorem 2.2 The number of k-permutations of $n$ distinguishable objects is

$$ {n \choose k} = {(n)_k \over k! } = {n! \over {k! (n - k)!}}$$

Theorem 2.4 Given $m$ distinguishable objects, there are $m^n$ ways to choose ith replacement an ordered sample of n objects.

Theorem 2.5 For $n$ repitions of a subexperiment with sample space $S_sub = {s_1, s_2, \cdots, s_m-1}$, the sample space $S$ of the sequential experiment has $m^n$ outcomes.

Theorem 2.6 The number of observation sequences for $n$ subexperiments with sample space $S = {0,1}$ with $0$ appearing $n_0$ times and $1$ appearing $n_1 = n - n_0$ times is $n \choose n_1$.

Theorem 2.7 For n reptitions of a subexperiment with sample space $S = {s_0, s_1, \cdots, s_m-1}$, the number of length $n = n_0 + n_1 + \cdots + n_{m-1}$ observation sequences with $s_i$ appearing $n_i$ times is

$$ {n \choose n_0, n_1, \cdots, n_{m-1}} = {n! \over {n_0! n_1! \cdots n_{m-1}!}} $$

Theorem 2.8 The probability of $n_0$ failures and $n_1$ successes in $n = n_0 + n_1$ independent trials is

$$ P[E_{n_0, n_1}] = {n \choose n_1} (1-p)^{n-n_1} p^n_1 = {n \choose n_0} (1-p)^{n_0} p^{n - n_0} $$

Theorem 2.9 A subexperiment has sample space $S = {s_0, s_1, \cdots, s_m-1}$ with $P[s_i] = p_i$ for $n = n_0 + n_1 + \cdots + n_{m-1}$ independent trials, the probability of $n_i$ occurrences of $s_i$, $i = 0, 1, \cdots, m-1$ is

$$ P[E_{n_0, n_1, \cdots, n_{m-1}}] = {n \choose n_0, n_1, \cdots, n_{m-1}} p_0^{n_0} p_1^{n_1} \cdots p_{m-1}^{n_{m-1}} $$

Definition 2.1 $n$ choose $k$ For an integer $n ≥ 0$, we define

$$ {n \choose k} = \begin{cases} {n! \over {k! (n - k)!}} & k = 0, 1, \dots, n, \\ 0 & \text{otherwise} \ \end{cases} $$

Definition 2.2 Multinomial coefficient $\space \text{For an integer n ≥ 0, we define }$

$${n \choose n_0, n_1, \dots, n_{m-1}} = {n! \over {n_0! n_1! \cdots n_{m-1}!}}$$

3. Discrete Random Variables

Theorem 3.1 For a discrete random variable X with PMF $P_X(x)$ and range $S_X: $

$\text{(a) For any x,} \space P_X(x) ≥ 0$

$\text{(b) } \sum_{x \in S_x} P_X(x) = 1$

$\text{(c) For any event} B \subset S_x, \space \text{The probability that X is in the set B is }$

$$P[B] = \sum_{x \in B} P_X(x)$$

Theorem 3.2 For any discrete random variable $X$ with range $S_x = { x_1, x_2, \dots }$ satisfying $x_1 ≤ x_2 ≤ \dots $,

$\text{(a) } F_X=(-\infty) = 0 \space \text{and} \space F_X(\infty) = 1 $

$\text{(b) For all } x' ≥ x, F_X(x') ≥ F_X(x) $

$\text{(c) For all } x' > x, F_X(x') > F_X(x) $

$\text{(d) } F_X(x) = F_X(x_i) \text{for all x such that } x_i ≤ x ≤ x_{i+1} $

Theorem 3.3 For all $b > a$, $F_X(b) - F_X(a) = P[a < X ≤ b] $

Theorem 3.4 The Bernoulli $(p)$ random variable $X$ has expected value $E[X] = p$

Theorem 3.5 The geometirc $(p)$ random variable $X$ has expect value $E[X] = 1/p$

Theorem 3.6

(a) For the binomial $(n, p)$ random variable $X$ of Definition 3.6

$$ E[X] = np \space $$

(b) For the Pascal $(k, p)$ random variable $X$ of Definition 3.7

$$ E[X] = k/p $$

(c) For the discrete uniform $(k, l)$ random variable $X$ of Definition 3.8

$$ E[X] = \frac{k + l}{2} $$

Theorem 3.8 Perfom $n$ Bernoulli trials. In each trial, let the probability of success be ${\alpha} / n$, where ${\alpha} > 0$ is a constant and $n >\alpha.$ Let the random variable $K_n$ be the number of successes in the $n$ trials. As $n \rightarrow \infty, P_{K_n}(k)$ converges to the PMF of a Poisson $(\alpha)$ random variable.

Theorem 3.9 For a discrete random variable $X$, the PMF of $Y = g(X)$ is

$$ P_Y(y) = \sum_{x: g(x) = y} P_X(x) $$

Theorem 3.10 Given a random variable $X$ with PMF $P_X(x),$ and the derived random variable $Y = g(x),$ the expected value of $Y$ is

$$ E[Y] = \mu_Y = \sum_{x \in S_x} g(x) P_X(x) $$

Theorem 3.11 For any random variable $X$

$$ E[X - \mu_{X}] = 0 $$

Theorem 3.12 For any random variable $X$

$$ E[aX + b] = aE[X] + b $$

Theorem 3.13 In the absence of observations, the minimum mean square error estimate random variable $X$ is

$$ \hat x = E[X] $$

Theorem 3.14

$$ Var[X] = E[X^2] - \mu^2_X = E[X^2] - (E[X])^2 $$

Theorem 3.15

$$ Var[aX + b] = a^2 Var[X] $$

Theorem 3.16

(a) If X is Bernoiulli $(p)$, then $Var[X] = p(1-p)$

(b) If X is geometric $(p)$, then $Var[X] = ({1-p})/{p^2}$

(c) If X is binomial $(n, p)$, then $Var[X] = np(1 - p)$

(d) If X is Pascal $(k, p)$, then $Var[X] = k(1 - p)/p^2$

(e) If X is Poisson $(\alpha)$ then $Var[X] = \alpha$

(f) If X is discrete uniform (k, l), then $Var[X] = (l - k)(l - k + 2)/12$

Definition 3.1 Random Variable

A random variable consists of an experiment with a probability measure $P[.]$ defined on a sample space S and a function that assigns a real number to each outcome in the sample spacce of the experiment.

Definition 3.2 Discrete Random Variable $X$ is a discrete random variable if the range of $X$ is a countable set.

$$ S_X = { x_1, x_2, \dots } $$

Definition 3.3 Probability Mass Function PMF The probability mass function (PMF) of a discrete random variable $X$ is a function that assigns a probability to each value in the range of $X$

$$ P_X(x) = P[X = x] $$

Definition 3.4 Bernoulii (p) Random Variable $X$ is a Bernoulli $(p)$ random variable if the PMF of X has the form

$$ {P_X(x)} = \begin{cases} {1-p} & x = 0 \\ {p} & x = 1 \\ 0 & \text{otherwise} \\ \end{cases} $$

$\text{where the parameter p is on the range } 0 < p < 1 $

Definition 3.5 Geometric (p) Random Variable $X$ is a geometric $(p)$ random variable if the PMF of $X$ has the form

$$ {P_X(x)} = \begin{cases} {p(1-p)^{x-1}} & x = 1, 2, 3, \dots \\ 0 & \text{otherwise} \\ \end{cases} $$

where the parameter p is on the range $0 < p < 1$

Definition 3.6 Binomial $\text{(n, p)}$ Random Variable X is a binomial (n, p) random variable if the PMF of X has the form

$$ P_X(s) = {n \choose x} p^x (1-p)^{n-x} $$

where $0 < p < 1$ and n is an integer such that $n ≥ 1$

Definition 3.7 Pascal $\text{(k, p)}$ Random Variable

$$ P_X(x) = {{x-1} \choose {k-1}} p^k (1-p)^{x-k} $$

where $0 < p < 1$ and k is an integer such that $k ≥ 1$

Definition 3.8 Discrete Uniform $\text{(k, l)}$ Random Variable $X$ is a discrete uniform $(k, l)$ random variable if the PMF of X has the form

$$ {P_X(x)} = \begin{cases} {1}/{(l - k + 1)} & x = k, k + 1, k + 2 , ... \space , l \\ 0 & \text{otherwise} \\ \end{cases} $$

where the parameters k and l are integers such that $k < l.$

Definition 3.9 Poisson $(\alpha)$ Random Variable $X$ is a Poisson $(\alpha)$ random variable if the PMF of X has the form

$$ P_X(x) = \begin{cases} {{\alpha^x e^{-\alpha}}/{x!}} \space & x = 0, 1, 2,\dots , \\ 0 & \space \text{otherwise} \\ \end{cases} $$

where the parameter $\alpha$ is in the range $\alpha > 0$

Definition 3.10 Cumulative Distribution Function (CDF) The cumulative distribution function (CDF) of a discrete random variable $X$ is a function that assigns a probability to each value in the range of $X$.

$$ F_X(x) = P[X \leq x] $$

Definition 3.11 Mode A mode of random variable $X$ is a number $x_{mod}$ satisfying $P_X(x_{mod}) ≥ P_X(x)$ for all $x$

Definition 3.12 Median A median $x_{med}$ of random variable $X$ is a number that satisfies

$$ P[X \leq x_{med}] = 1/2, \space{} \space{} P[X \geq x_{med}] = 1/2 $$

Definition 3.13 Expected Value The expected value of $X$ is

$$ E[X] = \mu_{X} = \sum_{x \in S_X} x P_X(x) $$

Definition 3.14 Derived Random Variable Each sample value y of a derived random variable $Y$ is a mathematical function $g(x)$ of a sample value $x$ of another random variable $X$. We adopt the notation $Y = g(X)$ to describe the relationship of the two random variables.

Definition 3.15 Variance The variance of random variable $X$ is

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

Definition 3.16 Standard Deviation The standard deviation of random variable $X$ is

$$ \sigma_X = \sqrt{Var[X]} $$

Definition 3.17 Moments For random variable $X$

(a) The nth moment is $E[X^n]$

(b) The nth central moment is $E[(X - \mu_X)^n]$

4. Continuous Random Variables

Theorem 4.1 For any random variable $X$,

(a) $F_X(-\infty) = 0$

(b) $F_X(\infty) = 1$

(c) $P[x_1 < X ≤ x_2] = F_X(x_2) - F_X(x_1)$

Theorem 4.2 For a continuous random variable $X$, with PMF $f_X(x)$,

(a) $f_X(x) ≥ 0 \text{ for all x, } $

(b) $f_X(x) = \int_{-\infty}^{x} f_X(u) \space du, $

(c) $\int_{-\infty}^{\infty} f_X(x) dx = 1$

Theorem 4.3

$$ P[x_1 < X ≤ x_2] = \int_{x_1}^{x_2} f_X(x) dx $$

Theorem 4.4 The expected value of a function, $g(X)$, of random variable $X$ is

$$ E[g(X)] = \int_{-\infty}^{\infty} g(x) f_X(x) dx $$

Theorem 4.5 For any random variable $X$,

(a) $E[X - \mu{X} ] = 0 $

(b) $E[aX + b] = aE[X] + b $

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

(d) $Var[aX + b] = a^2 Var[X] $

Theorem 4.6 If $X$ is a uniform $(a, b)$ random variable,

• The CDF of $X$ is

$$F_X(x) = \begin{cases} 0 & x < a \\ {(x - a)}/{(b - a)} & a ≤ x ≤ b \\ 1 & x > b \\ \end{cases} $$

• The expected value of $X$ is $E[X] = {(a + b)}/{2} $

• The variabce of $X$ is $Var[X] = {(b - a)^2}/{12} $

Theorem 4.7 Let $X$ be a uniform $(a, b)$ random variable, where $a$ and $b$ are both integers. Let $K = \lceil X \rceil$. Then $K$ is a discrete uniform $(a + 1, b)$ random variable.

Theorem 4.8 If $X$ is an exponential $(\lambda)$ random variable,

• The CDF of $X$ is

$$F_X(x) = \begin{cases} 1 - e^{-\lambda x} & x ≥ 0 \\ 0 & otherwise \\ \end{cases}$$

• The expected value of $X$ is $E[X] = {1}/{\lambda} $

• The variance of $X$ is $Var[X] = {1}/{\lambda^2} $

Theorem 4.9 If $X$ is an exponential $(\lambda)$ random variable, then $K = \lceil X \rceil$ is a geometric $(p)$ random variable with $p = 1 - e^{-\lambda}$

Theorem 4.10 If $X$ is an Erlang $(n, \lambda)$ random variable, then

$\text{(a) } E[X] = {n\over{(\lambda)}} $

$\text{(b) } Var[X] = {n\over{(\lambda)^2}} $

Theorem 4.11 Let $K_\alpha$ denote a Poisson $\alpha$ random variable. For any $x &gt; 0$, the CDF of an Erlang $(n, \lambda)$ random variable $X$ satisfies,

$$ F_X(x) = 1 - F_{K_\alpha}(n - 1) = \begin{cases} 1 - \sum_{k = 0}^{n - 1} \frac{(\lambda x)^k e^{-\lambda x}}{k!} & x ≥ n \\ 0 & otherwise \\ \end{cases} $$

Theorem 4.12 If $X$ is a Gaussian $(\mu, \sigma)$ random variable, then

$$ E[X] = \mu \space \space \space \space \space \space \space \space \space \space \space \space Var[X] = \sigma^2 $$

Theorem 4.13 If $X$ is a Gaussian $(\mu, \sigma), Y = aX + b$ is Gaussian $(a\mu + b, a\sigma)$

Theorem 4.14 If $X$ is a Gaussian $(\mu, \sigma)$ random variable, the CDF of $X$ is

$$ F_X(x) = \Phi \left( \frac{x - \mu}{\sigma} \right) $$

The probability that $X$ is in the interval $(a, b]$ is

$$ P[a < X ≤ b] = \Phi \left( \frac{b - \mu}{\sigma} \right) - \Phi \left( \frac{a - \mu}{\sigma} \right) $$

Theorem 4.15 $\space \space \Phi(-z) = 1 - \Phi(z)$

Theorem 4.16 For any continuous function g(x),

$$ \int_{-\infty}^{\infty} g(x) \delta(x - x_0) dx = g(x_0) $$

Theorem 4.17 $\int_{-\infty}^{x} \delta(v) dv = u(x) $

Theorem 4.18 For a random variable $X$, we have the folloing equivalent statements:

$\text{(a) } P[X = x_0] = q$

$\text{(b) } P[x_0] = q$

$\text{(c) } F_X(x_{0}^+) - F_X(x_{0}^-) = q$

$\text{(d) } f_x(x_0) = q \delta(0)$

Definition 4.1 Cumulative Distribution Function (CDF) The cumulative distribution function (CDF) of random variable $X$ is $F_X(x) = P[X ≤ x]$

Definition 4.2 Continuous Random Variable $X$ is a continuous random variable if the CDF F_X(x) is a continuous function.

Definition 4.3 Probability Density Function (PDF) The probability density function (PDF) of a continuous random variable $X$ is

$$f_X(x) = \frac{dF_X(x)}{dx}$$

Definition 4.4 Expected Value The expected value of a random variable $X$ is

$$E[X] = \int_{-\infty}^{\infty} x f_X(x) dx$$

Definition 4.5 Uniform Random Variable $X$ is a uniform $(a, b)$ random variable if the PDF of $X$ is $f_X(x)$, and where the parameter $\lambda &gt; 0$

$$f_X(x) = \begin{cases} {1}/{(b - a)} & \space a ≤ x ≤ b \\ 0 & \space otherwise \\ \end{cases} $$

Definition 4.6 Exponential Random Variable $X$ is an exponential (\lambda) random variable if the PDF of $X$ is $f_X(x)$, and where the parameter $\lambda &gt; 0$

$$f_x(x) = \begin{cases} \lambda e^{-\lambda x} & \space x ≥ 0 \\ 0 & \space otherwise \\ \end{cases} $$

Definition 4.7 Erlang Random Variable $X$ is an Erlang $(n, \lambda)$ random variable if the PDF of $X$ is $f_X(x)$ where the parameter $\lambda &gt; 0$, and the parameter $n ≥ 1$ is an integer.

$$f_X(x) = \begin{cases} \frac{\lambda^n x^{n - 1} e^{-\lambda x}}{(n - 1)!} & \space x ≥ n \\ 0 & \space otherwise \\ \end{cases} $$

Definition 4.8 Gaussian Random Variable $X$ is a Gaussian $(\mu, \sigma)$ random variable if the PDF of $X$ is $f_X(x)$ where the parameter $\mu$ can be any real number and the parameter $\sigma &gt; 0$

$$\begin{align} f_X(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-{(x - \mu)^2}/{2\sigma^2}} \end{align}$$

Definition 4.9 Standard Normal Random Variable The standard normal random variable $Z$ is the Gaussian $(0, 1)$ random variable.

Definition 4.10 Standard Normal CDF The CDF of the standard normal random variable $Z$ is

$$\Phi(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z} e^{-{u^2}/{2}} du$$

Definition 4.11 Standard Normal Complementary CDF The standard normal complementary CDF is

$$ Q(z) = P[Z > z] = \frac{1}{\sqrt{2\pi}} \int_{z}^{\infty} e^{-{u^2}/{2}} du = 1 - \Phi(z)$$

Definition 4.12 Unit Impluse (Delta) Function Let

$$\delta(x) = \begin{cases} 1/{\epsilon} & -\epsilon/2 ≤ x ≤ \epsilon/2 \\ 0 & \space otherwise \\ \end{cases} $$

The unit impulse function

$$\delta(x) = \lim_{x \to 0} d_{\epsilon}(x)$$

Definition 4.13 Unit Step Function The unit step function is

$$u(x) = \begin{cases} 1 & x < 0 \\ 0 & x ≥ 0 \\ \end{cases} $$

Definition 4.14 Mixed Random Variable $X$ is a mixed random variable if and only if $F_X(x)$ contains both impluses and nonzero, finite values.