Random Variables

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What is a random variable?

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

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What is the difference between discrete and continuous random variables?

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Discrete random variables take on a countable number of distinct values, while continuous random variables can take on any value within a given range.

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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 (the number of trials) and p (the probability of success on each trial).

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What is the expected value of a random variable?

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The expected value is the long-term average or mean of a random variable, calculated as the sum of all possible values, each multiplied by its probability.

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How do you calculate the expected value for a discrete random variable?

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E(X) = Σ [x * P(x)], where x is a value of the random variable and P(x) is the probability of x.

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What is the formula for the probability of getting exactly k successes in a binomial distribution?

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P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is the binomial coefficient.

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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.

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What is the mean of a probability distribution?

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The mean is the average value of the random variable, weighted by the probabilities of each outcome.

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What is the variance of a random variable?

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Variance measures the spread of a random variable's values around the mean, calculated as E[(X - μ)²].