Lecture 14 Probability Distributions Discrete Random Variables_NEW2

Probability Distributions

Combination of frequency distributions, descriptive statistics, and probability to describe probable outcomes.

Random Variables

Variables (e.g., 𝑋) that take different values based on random phenomena, representing numerical outcomes determined by chance.

Probability Distribution Function

A graph, table, or formula that assigns probabilities to each value of a random variable, known as probability mass function for discrete distributions.

Requirements for a Probability Distribution

Individual probabilities must be between 0 and 1, and the sum of all probabilities must equal 1.

Discrete Random Variables

Variables that have a finite or countable number of values, taking certain numerical values without intermediate values.

Expected Value

Mean of a discrete random variable 𝑋, calculated as μ = E(X) = ∑ -x P(x).

Variance

Measure of the spread of a random variable 𝑋, calculated as σ² = E(X²) - μ² = ∑ -x² P(x) - μ².

Binomial Distribution

A distribution characterized by a fixed number of identical trials, two possible outcomes, constant probability of success, and independent trials.

Binomial Coefficient

Number of ways to choose 𝑘 objects from 𝑛 objects, denoted as C(n, k) or nCk.

Factorials

Defined as x! = 1 × 2 × ... × x, used in calculations involving combinations.

Binomial PDF

Probability distribution function for a binomial random variable, expressed as P(X = x) = C(n, x) p^x (1 - p)^{n - x}.

Poisson Distribution

A distribution that models counts data, applicable for events occurring in a given time or space interval.

Poisson PDF

Probability of observing 𝑘 events given by P(X = k) = (e^{−λ} λ^k) / k!.

Parameters of Poisson Distribution

Mean (λ) is equal to variance, and standard deviation is √λ.

Language of Probability

Definitions of phrases like "at least 𝑥" and "no more than 𝑥" used in probability contexts.

What is the combination of methods in probability distributions?

Merges frequency distributions, descriptive statistics, and probability.

What does a probability distribution describe?

Probable outcomes rather than actual occurrences.

What are random variables?

Variables that take different values based on random phenomena.

What does the random variable 𝑋 represent in the example of tossing 3 coins?

The number of heads.

What is a probability distribution function?

A graph, table, or formula that assigns probabilities to each value of a random variable.

What are the requirements for a probability distribution?

Individual probabilities must be between 0 and 1, and the sum of all probabilities must equal 1.

How are discrete random variables defined?

They have a finite or countable number of values.

Give an example of a discrete random variable.

𝑋 = Number of Heads from a coin toss.

What does the population in probability distributions refer to?

All possible outcomes for a random variable.

How is the expected value of a discrete random variable calculated?

μ = E(X) = ∑ -x P(x).

What are the characteristics of a binomial distribution?

Fixed number of trials, two possible outcomes, constant probability of success, and independent trials.

What are the parameters of a binomial distribution?

Number of successes 𝑋 out of 𝑛 trials, with parameters 𝑛 and 𝑝.

What is a binomial coefficient?

The number of ways to choose 𝑘 objects from 𝑛 objects, denoted as C(n, k).

How is a factorial defined?

x! = 1 × 2 × ... × x.

What is the binomial PDF formula?

P(X = x) = C(n, x) p^x (1 - p)^{n - x}.

What does the notation for binomial distributions include?

𝑆:Success, 𝐹:Failure, 𝑛:Number of trials, 𝑝:Probability of success, 𝑞:Probability of failure.

What is the Poisson distribution used for?

Modeling counts data for events occurring in a given time or space interval.

What is the Poisson PDF formula?

P(X = k) = (e^{-\lambda} λ^k) / k!.

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