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 are the key characteristics of a binomial distribution?

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1. Fixed number of trials (n). 2. Two possible outcomes (success or failure). 3. Constant probability of success (p) for each trial. 4. Trials are independent.

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What are the key characteristics of a geometric distribution?

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1. Trials continue until the first success. 2. Each trial has two outcomes (success or failure). 3. The probability of success (p) is constant. 4. Trials are independent.

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How do you calculate the expected value of a binomial distribution?

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The expected value (mean) of a binomial distribution is calculated using the formula: 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 expected value of a geometric distribution?

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The expected value (mean) of a geometric distribution is calculated using the formula: E(X) = 1/p, where p is the probability of success.

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What is the formula for calculating 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 given by: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), 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 probability of the first success on the k-th trial in a geometric distribution?

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The probability of the first success on the k-th trial in a geometric distribution is given by: P(X = k) = (1-p)^(k-1) * p, where p is the probability of success.

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In a binomial distribution, what does the term 'n choose k' represent?

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'n choose k' (denoted as C(n, k) or nCk) represents the number of ways to choose k successes from n trials and is calculated as: C(n, k) = n! / (k!(n-k)!).

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What is the significance of the Central Limit Theorem in relation to binomial distributions?

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The Central Limit Theorem states that as the number of trials (n) increases, the binomial distribution approaches a normal distribution, allowing for the use of normal approximation for large n.