6.3 Flashcard Binomial and Geometric Distributions Make-Up

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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 expected value in a binomial distribution?

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The expected value (mean) in a binomial distribution is calculated as E(X) = n * p, where n is the number of trials and p is the probability of success.

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

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It means that the variable represents the number of trials until the first success occurs, with each trial being independent and having the same probability of success.

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What is the standard deviation in a binomial distribution?

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The standard deviation in a binomial distribution is calculated as SD = sqrt(n * p * (1 - p)).

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What is the mean of a geometric distribution?

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The mean of a geometric distribution is calculated as 1/p, where p is the probability of success.

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What is the variance in a geometric distribution?

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The variance in a geometric distribution is calculated as (1 - p) / p^2, where p is the probability of success.

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How do you interpret the expected number of successes in a sample?

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The expected number of successes in a sample is the average number of successes you would expect if you repeated the sampling process many times.

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What is the relationship between sample size and expected value in binomial distributions?

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As the sample size increases, the expected value (mean) also increases, as it is directly proportional to the number of trials.