Random Variables and Probability Distributions

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What is a random variable?

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A random variable is a numerical outcome of a random phenomenon, which can be discrete (taking specific values) or continuous (taking any value within a range).

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What is a probability distribution?

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A probability distribution describes how the probabilities are distributed over the values of the random variable. It can be represented as a table, graph, or mathematical function.

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What is a binomial distribution?

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A binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

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What are the parameters of a binomial distribution?

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The parameters are n (number of trials) and p (probability of success on each trial).

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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.

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What is the shape of a geometric distribution?

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The shape of a geometric distribution is always skewed right.

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What does 'n choose k' mean?

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'n choose k' or C(n, k) represents the number of combinations of k items that can be selected from a total of n items.

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What is the formula for combinations?

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The formula for combinations is C(n, k) = n! / (k!(n-k)!), where n! is the factorial of n.

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What is the difference between permutations and combinations?

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Permutations consider the order of selection, while combinations do not.

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What is a binomial experiment?

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A binomial experiment consists of a fixed number of trials, each trial has two possible outcomes (success or failure), and the trials are independent.