AP Stats Unit 4 (pt 2) Random Variables (w/out Geo)

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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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CCSS.HSS.MD.A.2

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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 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's possible values, weighted by their probabilities.

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CCSS.HSS.MD.A.2

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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 that value.

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CCSS.HSS.MD.A.2

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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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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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What is the standard deviation of a random variable?

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The standard deviation measures the dispersion or spread of a set of values around the mean of the random variable.

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How do you find the standard deviation of the sum of two independent random variables?

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If X and Y are independent random variables, then the standard deviation of their sum is given by: SD(X + Y) = √(SD(X)² + SD(Y)²).

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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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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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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) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, p is the probability of success, and k is the number of successes.

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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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What is the Central Limit Theorem?

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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.