Discrete Random Variables

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What is a discrete random variable?

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

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How do you calculate the mean of a discrete random variable?

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To calculate the mean, multiply each value of the random variable by its probability and sum all the products.

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What is the formula for variance of a discrete random variable?

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Variance = Σ [ (x - μ)² * P(x) ], where x is each value, μ is the mean, and P(x) is the probability of x.

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What is the standard deviation of a discrete random variable?

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The standard deviation is the square root of the variance, providing a measure of the spread of the values.

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What is a probability distribution?

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A probability distribution describes how probabilities are distributed over the values of a random variable.

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What is the difference between a probability mass function (PMF) and a cumulative distribution function (CDF)?

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PMF gives the probability that a discrete random variable is exactly equal to a value, while CDF gives the probability that the variable is less than or equal to a value.

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What does it mean for a distribution to be a valid probability distribution?

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A valid probability distribution must satisfy two conditions: all probabilities must be between 0 and 1, and the sum of all probabilities must equal 1.

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How do you find the probability of at least k successes in a binomial distribution?

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Use the binomial probability formula and sum the probabilities from k to n, or use the complement rule.

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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 value of repetitions of the experiment it represents, calculated as E(X) = Σ [ x * P(x) ].

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What is the significance of the central limit theorem in statistics?

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The central limit theorem states that the distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the original distribution.