Bayes Rule

\[\boxed{P(A|B) = \frac{P(B|A)P(A)}{P(B)}}\]

Where:

• \(P(A)\) = prior;
• \(P(B \mid A)\) = likelihood;
• \(P(A \mid B)\) = posterior;

Random Variables

Probability Density Function Formula

The cumulative distribution function (CDF) is a function that gives the probability that a random variable takes on a value less than or equal to a given value. It is denoted as \(F(x)\). The CDF is defined as:

\[\boxed{F(x) = P(X \leq x)}\]

Where \(X\) is the random variable.

Cumulative Distribution Function as an Integral

\[\boxed{F(x) = \int_{-\infty}^{x} \rho(y) dy}\]

Where \(\rho(y)\) is the probability density function (PDF) of the random variable \(X\).

Joint Probability Distributions

The joint probability density function (PDF) is a function that describes the probability of multiple random variables taking on specific values simultaneously. It is denoted as \(f(x_1, x_2, ..., x_n)\), where \(x_1, x_2, ..., x_n\) are the values of the random variables.

Properties:

• Non-negativity: \(f(x_1, x_2) \geq 0\) for all \(x_1, x_2\).
• Normalization: \(\int\int f(x_1, x_2) dx_1 dx_2 = 1\), where the integral is taken over the entire range of each random variable.

The joint cumulative distribution function (CDF) is a function that gives the probability that multiple random variables take on values less than or equal to given values simultaneously. It is denoted as \(F(x_1, x_2, ..., x_n)\), where \(x_1, x_2, ..., x_n\) are the values of the random variables.

\[F(x_1, x_2, ..., x_n) = P(X_1 \leq x_1, X_2 \leq x_2, ..., X_n \leq x_n)\]

For a joint distribution of multiple random variables \(X_1, X_2, ..., X_n\), the joint CDF \(F(x_1, x_2, ..., x_n)\) is given by:

\[F(x_1, x_2, ..., x_n) = \int...\int f(x_1, x_2, ..., x_n) dx_1 dx_2 ... dx_n\]

This integral formula allows us to calculate the probability that multiple random variables take on values less than or equal to given values simultaneously.

Probability Distributions

Binomial Distribution

The binomial distribution gives the probability of \(k\) number of successes in \(n\) independent trials, where each trial has a probability \(p\) of success.

\[\boxed{P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}}\]

Characteristics:

• Mean: \(np\)
• Variance: \(\sigma^2 = np(1-p)\)

Examples:

• Coin flips
• Binary outcome events

Poisson Distribution

The Poisson distribution gives the probability of the number of events occurring within a fixed interval where the known, constant rate of each event’s occurrence is \(\lambda\).

\[\boxed{P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}}\]

Characteristics:

• Mean: \(\lambda\)
• Variance: \(\lambda\)

Examples:

• Assessing counts over a continuous interval
  • Number of visits to a website in a certain period of time

Continuous Probability Distributions

Uniform Distribution

The uniform distribution implies that all outcomes are equally likely within a given range.

PDF: \(\boxed{f(x) = \begin{cases} \frac{1}{b-a} & \text{if } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}}\)

Characteristics:

• Mean: \(\frac{a+b}{2}\)
• Variance: \(\frac{(b-a)^2}{12}\)

Examples:

• Rolling a fair die
• Sampling (random number generation)

Exponential Distribution

The exponential distribution represents the probability of the interval length between events of a Poisson process having a set rate parameter of \(\lambda\).

PDF: \(\boxed{f(x) = \begin{cases} \lambda e^{-\lambda x} & \text{if } x \geq 0 \\ 0 & \text{otherwise} \end{cases}}\)

Characteristics:

• Mean: \(\frac{1}{\lambda}\)
• Variance: \(\frac{1}{\lambda^2}\)
• Memoryless Property: \(P(X > s+t \mid X > s) = P(X > t)\)

Normal Distribution

The normal distribution is characterized by a bell-shaped curve where the mean, median, and mode are all equal.

PDF: \(\boxed{f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}}\)

Characteristics:

• Mean: \(\mu\)
• Variance: \(\sigma^2\)

Markov Chains

• probability of being in a particular state is only dependent on the previous state
• state space: set of all possible states
• transition matrix: \(P = [p_{ij}]\) where \(p_{ij}\) is the probability of transitioning from state \(i\) to state \(j\)

\[P = \begin{bmatrix} p_{11} & p_{12} & \dots & p_{1n} \\ p_{21} & p_{22} & \dots & p_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ p_{n1} & p_{n2} & \dots & p_{nn} \\ \end{bmatrix}\]

Each row of the matrix represents the probabilities of transitioning from a particular state to all other states. The sum of probabilities in each row should be equal to 1.

• recurrent state: a state that can be reached again from itself
• transient state: a state that cannot be reached again from itself stationary distribution for markov chain: \(\pi = \pi P\)
  • \(P\) is the transition matrix