Discrete Probability Distributions - Review

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What is a discrete random variable?

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A discrete random variable is a variable that can take on a countable number of distinct values, often representing counts or whole numbers.

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What is the formula for 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)), where n is the number of trials and p is the probability of success.

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What is the probability of getting exactly k successes in n trials in a binomial distribution?

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The probability of getting exactly k successes in n trials is given by the formula: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k), where (n choose k) = n! / (k!(n-k)!) is the binomial coefficient.

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What are the assumptions of a binomial distribution?

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1. Fixed number of trials (n). 2. Each trial is independent. 3. Each trial has only two outcomes (success or failure). 4. The probability of success (p) is constant for each trial.

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What is 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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How do you calculate the probability of at least one success in a binomial distribution?

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To find the probability of at least one success, use the complement rule: P(at least 1 success) = 1 - P(no successes) = 1 - (1 - p)^n.

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What is the Poisson distribution used for?

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The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space, given a known average rate of occurrence.

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What is the mean of a Poisson distribution?

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The mean (λ) of a Poisson distribution is equal to the average number of occurrences in the given interval.

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How do you find the probability of k events in a Poisson distribution?

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The probability of observing k events in a Poisson distribution is given by the formula: P(X = k) = (λ^k * e^(-λ)) / k!, where e is Euler's number.

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What is the difference between discrete and continuous random variables?

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Discrete random variables can take on a countable number of values, while continuous random variables can take on an infinite number of values within a given range.