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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What is the probability mass function (PMF)?

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The probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value.

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What is the expected value of a discrete 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=x)] for all possible values x.

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What does it mean for probabilities to sum to 1?

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For a discrete random variable, the sum of the probabilities of all possible outcomes must equal 1, indicating that one of the outcomes must occur.

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What is the variance of a discrete random variable?

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Variance measures the spread of a set of values, calculated as Var(X) = E(X^2) - (E(X))^2.

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What is the difference between discrete and continuous data?

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Discrete data consists of distinct, separate values (e.g., number of students), while continuous data can take any value within a range (e.g., height, weight).

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How do you calculate P(X < k) for a discrete random variable?

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P(X < k) is calculated by summing the probabilities of all outcomes less than k, i.e., P(X < k) = Σ P(X=x) for all x < k.

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What is a cumulative distribution function (CDF)?

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The cumulative distribution function (CDF) of a discrete random variable gives the probability that the variable is less than or equal to a certain value, defined as F(x) = P(X ≤ x).

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What is the significance of the mode in a discrete distribution?

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The mode is the value that appears most frequently in a discrete distribution, indicating the most likely outcome.

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What is the range of a discrete random variable?

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The range of a discrete random variable is the set of all possible values it can take.