# Discrete distributions_bern_binom_poisson_geom_imported Flashcards

What is the Bernoulli distribution?

The Bernoulli distribution models a random experiment with two outcomes: success (1) and failure (0).

How is the Bernoulli distribution defined mathematically?

The PMF of the Bernoulli distribution is: P(X = x) = p^x * (1 - p)^(1 - x), where x is 0 or 1, and p is the probability of success.

What are the mean and variance of the Bernoulli distribution?

Mean: E(X) = p, Variance: Var(X) = p * (1 - p), where x is 0 or 1, and p is the probability of success.

Can you provide an example of a Bernoulli distribution using a fair coin flip?

Example: Fair coin flip - P(X = 1) = 0.5, P(X = 0) = 0.5

How about an example with a biased coin?

Example: Biased coin - P(X = 1) = 0.3, P(X = 0) = 0.7

Can you provide a real-world application example of the Bernoulli distribution?

Example: Online ad CTR - P(X = 1) = 0.1, P(X = 0) = 0.9 (10% chance of clicking, 90% chance of not clicking).

What is the Probability Mass Function (PMF) for the Bernoulli distribution?

The PMF of the Bernoulli distribution is given by: P(X = x) = p^x * (1 - p)^(1 - x), where x is the value (usually 0 or 1) and p is the probability of success.

Could you provide an example of PMF for a biased coin with p = 0.7?

For a biased coin with probability of heads p = 0.7, the PMF is: P(X = 1) = 0.7, P(X = 0) = 0.3.

What is the Cumulative Distribution Function (CDF) for the Bernoulli distribution?

The CDF of the Bernoulli distribution is: CDF(X ≤ x) = 1 - (1 - p)^x, where x is the value and p is the probability of success.

Can you provide an example of CDF for the biased coin with p = 0.7?

For the biased coin with p = 0.7, the CDF for X ≤ 1 is calculated as: CDF(X ≤ 1) = 0.7.

How is the Expectation (Mean) calculated for the Bernoulli distribution?

The expectation (mean) of a Bernoulli-distributed random variable is given by: E(X) = p, where p is the probability of success.

What’s the expectation for the biased coin with p = 0.7?

For the biased coin with p = 0.7, the expectation is: E(X) = 0.7.

What’s the formula for Variance in the Bernoulli distribution?

The variance of a Bernoulli-distributed random variable is: Var(X) = p * (1 - p), where p is the probability of success.

How about an example of Variance for the biased coin with p = 0.7?

For the biased coin with p = 0.7, the variance is calculated as: Var(X) = 0.7 * (1 - 0.7) = 0.21.

What is the expectation in probability and statistics?

The expectation, often referred to as the “expected value” or “mean,” is a fundamental concept in probability and statistics. It represents the average or central value of a random variable, considering the probabilities associated with each possible outcome.

How is the expectation calculated for a discrete random variable?

For a discrete random variable X with values x₁, x₂, …, xₙ and corresponding probabilities p(x₁), p(x₂), …, p(xₙ), the expectation E(X) is calculated by multiplying each value by its probability and summing the results: E(X) = x₁ * p(x₁) + x₂ * p(x₂) + … + xₙ * p(xₙ)

Can you provide an example of calculating expectation?

Certainly. For a fair six-sided die roll with equal probabilities, the expectation of the outcome (X) is calculated as: E(X) = 1 * (1/6) + 2 * (1/6) + 3 * (1/6) + 4 * (1/6) + 5 * (1/6) + 6 * (1/6) = 3.5

How does the expectation relate to the concept of an average?

The expectation is akin to an average, but it considers not only the values themselves but also the likelihood of each value occurring. In essence, the expectation is a weighted average, where the weights are the probabilities of each outcome.

What is the significance of the expectation in probability and statistics?

The expectation provides valuable information about the center or typical value of a random variable’s distribution. It helps in making informed decisions and predictions based on probabilistic outcomes. Understanding the expectation is crucial for assessing risks, estimating future values, and designing strategies.

What is variance in probability and statistics?

Variance is a statistical measure that quantifies the spread or dispersion of data points or the variability of a random variable. It indicates how much individual data points deviate from the mean or expected value.

How is variance calculated for a set of data?

For a set of data points x₁, x₂, …, xₙ with mean µ, the variance is calculated as the average of the squared differences between each data point and the mean.

What is the formula for calculating variance of a discrete random variable?

For a discrete random variable X with values x₁, x₂, …, xₙ and corresponding probabilities p(x₁), p(x₂), …, p(xₙ), the variance Var(X) is calculated using the squared differences between each value and the expectation, weighted by their probabilities.

Can you provide an example of calculating variance for a set of data?

Certainly. For the data set {3, 5, 7, 10, 12} with a mean of 7.4, the variance is calculated as approximately 10.96.

How does variance relate to the concept of data spread?

Variance quantifies how much data points spread out around the mean. A larger variance indicates greater spread or variability, while smaller variance indicates less spread.

What is the significance of variance in statistical analysis?

Variance is crucial for comparing data spread, assessing reliability, making predictions, and understanding outcome distribution. It is a key element in statistical methods like hypothesis testing, regression, and quality control.

Is the expectation related to the population or sample in statistics?

The concept of expectation is related to both population and sample in statistics. It is referred to as the “population mean” or “population expected value” when considering an entire population, and as the “sample mean” when dealing with a subset of data from a larger population.

What is the population expectation (population mean)?

In the context of a population, the expectation is commonly referred to as the “population mean” or “population expected value.” It represents the average value of the entire population, considering the probabilities or frequencies of each possible value.