6.3 Flashcard Binomial and Geometric Distributions

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What is a binomial distribution?

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A binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states and is defined by two parameters: the number of trials (n) and the probability of success (p).

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What is a geometric distribution?

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A geometric distribution models the number of trials needed to get the first success in a series of independent Bernoulli trials, where each trial has the same probability of success.

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What is the formula for calculating the mean of a binomial distribution?

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The mean (μ) of a binomial distribution is calculated using the formula: μ = n * p, where n is the number of trials and p is the probability of success.

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What is the formula for calculating the standard deviation of a binomial distribution?

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The standard deviation (σ) of a binomial distribution is calculated using the formula: σ = √(n * p * (1 - p)).

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How do you interpret the standard deviation in the context of a binomial distribution?

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The standard deviation indicates the average distance of the observed values from the mean, providing insight into the variability of the distribution.

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What is the probability of getting at least one success in a geometric distribution?

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The probability of getting at least one success in a geometric distribution is calculated as: P(X ≥ 1) = 1 - (1 - p)^k, where p is the probability of success and k is the number of trials.

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What does it mean if a random variable follows a binomial distribution?

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It means that the random variable represents the number of successes in a fixed number of independent trials, each with the same probability of success.

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What does it mean if a random variable follows a geometric distribution?

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It means that the random variable represents the number of trials until the first success occurs in a series of independent trials.

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How do you calculate the probability of exactly k successes in a binomial distribution?

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The probability of exactly k successes in a binomial distribution is calculated using the formula: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k).

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What is the significance of the 'success' and 'failure' outcomes in binomial and geometric distributions?

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In both distributions, 'success' refers to the outcome of interest, while 'failure' refers to the complementary outcome. The probabilities of these outcomes are used to calculate various probabilities.