Random Variables

• Definition: A function mapping sample space Ω to real line R.

• Example: Let X be number of heads in 4 tosses of a coin. Possible values: 0, 1, 2, 3, 4.

Probability Distributions

• Types: Discrete (finite/countably infinite values) & Continuous.

• Conditions for probability function of X:

  1. f_X(x) \geq 0, \forall x

  2. \sumx fX(x) = 1

Bernoulli Distribution

• Two outcomes: success (1) and failure (0).

• Probability function:

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

  • Parameter: p

Binomial Distribution

• Sum of n independent Bernoulli trials:

• Probability function:

  • f_X(x) = \binom{n}{x} p^x (1-p)^{n-x}, \quad x = 0, 1, \ldots, n

Geometric Distribution

• Number of trials up to first success:

• Probability function:

  • f_X(x) = (1-p)^{x-1} p, \quad x = 1, 2, \ldots

Negative Binomial Distribution

• Number of trials to achieve k successes:

• Probability function:

  • f_X(x) = \binom{x-1}{k-1} p^k (1-p)^{x-k}, \quad x = k, k+1, \ldots

Hypergeometric Distribution

• Sampling without replacement:

• Probability function:

  • f_X(x) = \frac{\binom{k}{x} \binom{N-k}{n-x}}{\binom{N}{n}}, \text{where } 0 \leq x \leq \min(k,n)

Poisson Distribution

• Model for count of events in fixed interval:

• Probability function:

  • f_X(x) = \frac{e^{-\lambda} \lambda^x}{x!}, \quad x = 0, 1, 2, \ldots

Poisson Process

• A model for random events over time:

  • Conditions include constant arrival rate and independence of events.

• Probability function for number of events in interval:

  • P(X(t) = x) = \frac{(\lambda t)^x e^{-\lambda t}}{x!} for each time t.

Multinomial Distribution

• Generalization of binomial distribution with k outcomes:

• Introduced by parameters n (trials) and $ p1, p2, \ldots, p_k $:

  • P(Y1 = y1, Y2 = y2, \ldots, Yk = yk) = \frac{n!}{y1! y2! \ldots yk!} p1^{y1} p2^{y2} \ldots pk^{y_k}

Random Variables

Definition: A random variable is a function that maps each outcome of a probabilistic experiment (the sample space, denoted as Ω) to a corresponding real number on the real line (denoted as R). This mapping allows us to quantify uncertain outcomes in a systematic way, facilitating the application of probability theory to various fields such as statistics, finance, and natural sciences.

Example: Consider a simple experiment where a fair coin is tossed four times. Let X be the random variable representing the number of heads observed in these tosses. The possible values that X can take are 0 (no heads), 1 (one head), 2 (two heads), 3 (three heads), and 4 (four heads). Each of these outcomes can be analyzed for its probability of occurrence, which provides insight into the distribution of X.

Probability Distributions

Probability distributions describe how probabilities are assigned to the possible values of a random variable. There are two main types of probability distributions: discrete and continuous.

• Discrete Probability Distributions: These are used when a random variable can take on a finite or countably infinite set of values. Examples include the Binomial distribution and the Poisson distribution.

• Continuous Probability Distributions: These are used when a random variable can take on any value in a given range, such as the Normal distribution.

Conditions for the probability function of a random variable X include:

1. f_X(x) \geq 0, \forall x (probability must be non-negative)

2. \sumx fX(x) = 1 (the sum of probabilities over all possible outcomes must equal one)

Bernoulli Distribution

The Bernoulli distribution describes a random experiment with exactly two possible outcomes: success (1) and failure (0). It is a fundamental building block for more complex distributions.

Probability function:
f_X(x) = \begin{cases} 1-p & \text{if } x=0 \ p & \text{if } x=1 \ 0 & \text{otherwise} \end{cases}

Parameter: The single parameter p represents the probability of success in a Bernoulli trial, while the probability of failure is represented as 1-p.

Binomial Distribution

The Binomial distribution extends the Bernoulli distribution to situations where multiple independent Bernoulli trials are conducted. It represents the number of successes in n independent trials.

Probability function:
f_X(x) = \binom{n}{x} p^x (1-p)^{n-x}, \quad x = 0, 1, \ldots, n

Here, \binom{n}{x} is the binomial coefficient representing the number of ways to choose x successes from n trials.

Geometric Distribution

The Geometric distribution measures the number of trials needed to achieve the first success in a series of independent Bernoulli trials. It is especially useful in scenarios involving repeated attempts of an experiment until success is achieved.

Probability function:
f_X(x) = (1-p)^{x-1} p, \quad x = 1, 2, \ldots

Negative Binomial Distribution

The Negative Binomial distribution generalizes the Geometric distribution to model the number of trials required to achieve k successes rather than just one. This distribution proves relevant in various practical applications, such as in quality control and reliability testing.

Probability function:
f_X(x) = \binom{x-1}{k-1} p^k (1-p)^{x-k}, \quad x = k, k+1, \ldots

Hypergeometric Distribution

The Hypergeometric distribution applies in scenarios of sampling without replacement. This distribution is crucial for understanding the probability of particular outcomes when a finite population is sampled multiple times.

Probability function:
f_X(x) = \frac{\binom{k}{x} \binom{N-k}{n-x}}{\binom{N}{n}}, \text{where } 0 \leq x \leq \min(k,n)

In this expression, N is the population size, k is the number of successes in the population, and n is the size of the drawn sample.

Poisson Distribution

The Poisson distribution models the number of events that occur in a fixed interval of time or space, making it particularly useful in fields such as telecommunications and traffic flow analysis.

Probability function:
f_X(x) = \frac{e^{-\lambda} \lambda^x}{x!}, \quad x = 0, 1, 2, \ldots

Here, \lambda is the average rate (mean) of occurrence of the events in the specified interval.

Poisson Process

A Poisson process describes a model for a series of events occurring randomly and independently over time. Key conditions include a constant average rate of occurrence and the independence of events.

Probability function for the number of events in a specified time interval:
P(X(t) = x) = \frac{(\lambda t)^x e^{-\lambda t}}{x!} for each time t.

Multinomial Distribution

The Multinomial distribution generalizes the Binomial distribution to cases where there are more than two possible outcomes for each trial. It is widely used in various fields, including economics, biology, and survey analysis.

It is introduced by parameters n (number of trials) and $ p1, p2, \ldots, pk $ (the probabilities of each outcome): P(Y1 = y1, Y2 = y2, \ldots, Yk = yk) = \frac{n!}{y1! y2! \ldots yk!} p1^{y1} p2^{y2} \ldots pk^{yk}