Stats 1 Week 11

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Probability Distributions

Bernoulli Trials

A Bernoulli trial is a random experiment with exactly two possible outcomes: "success" and "failure". The probability of success is denoted by ( p ), and the probability of failure is ( 1 - p ). Examples include flipping a coin (heads or tails) and checking if a light bulb is functional (working or not).

Non-Bernoulli Trials

Non-Bernoulli trials are experiments that do not fit the criteria of Bernoulli trials. They may have more than two outcomes or the probability of success may change from trial to trial.

Bernoulli Random Variable

A Bernoulli random variable \( X \) is a random variable that takes the value 1 with probability \( p \) (success) and 0 with probability \( 1 - p \) (failure). It is denoted as \( X \sim \text{Bernoulli}(p) \).

Expectation and Variance of Bernoulli Random Variable

Expectation (Mean): \( E(X) = p \)

Variance: \( \text{Var}(X) = p(1 - p) \)

Independent and Identically Distributed (i.i.d.) Bernoulli Trials

Independent and identically distributed (i.i.d.) Bernoulli trials are a sequence of Bernoulli trials where each trial is independent of the others, and each trial has the same probability of success \( p \).

Non-Independent Trials

Non-independent trials are those where the outcome of one trial affects the outcome of another. For example, drawing cards from a deck without replacement.

Binomial Random Variable

A binomial random variable \( X \) represents the number of successes in \( n \) independent Bernoulli trials with the same probability of success \( p \). It is denoted as \( X \sim \text{Binomial}(n, p) \).

Notation of Binomial Random Variable

The binomial random variable \( X \) is written as \( X \sim \text{Binomial}(n, p) \), where:

• \( n \) is the number of trials
• \( p \) is the probability of success in each trial

When Binomial Distribution is Used

The binomial distribution is used when we are interested in the number of successes in a fixed number of independent trials, each with the same probability of success. Examples include the number of heads in multiple coin tosses and the number of defective items in a batch.

Uses in Real Life

• Quality control in manufacturing
• Survey results (e.g., number of people who prefer a product)
• Medical trials (e.g., number of patients who recover after treatment)

Probability Mass Function (PMF) of Binomial Distribution

The PMF of a binomial random variable \( X \) is given by:

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

where \( k \) is the number of successes, \( \binom{n}{k} \) is the binomial coefficient.

Hypergeometric Distribution

The hypergeometric distribution describes the probability of \( k \) successes in \( n \) draws from a finite population of size \( N \) containing \( K \) successes, without replacement. It is used when the draws are dependent.

Example: I have added one example in notes in hypergeometric, go through it. (Question won't be asked from this distribution, still keep basic in mind.)

Expectation and Variance of Hypergeometric Distribution

Expectation (Mean): \( E(X) = n \frac{K}{N} \)

Variance: \( \text{Var}(X) = n \frac{K}{N} \left(1 - \frac{K}{N}\right) \frac{N - n}{N - 1} \)

Poisson Distribution (Very Important)

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given the events occur with a known constant mean rate and independently of the time since the last event.

When Poisson Distribution is Used

The Poisson distribution is used for modeling the number of events in a fixed interval of time or space. Examples include the number of phone calls received by a call center in an hour and the number of decay events per unit time from a radioactive source.

Expectation and Variance of Poisson Distribution

Expectation (Mean): \( E(X) = \lambda \)

Variance: \( \text{Var}(X) = \lambda \)

Probability Mass Function (PMF) of Poisson Distribution

The PMF of a Poisson random variable \( X \) is given by:

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

where \( \lambda \) is the average rate of occurrence, and \( k \) is the number of occurrences.