Random Variables Review

Front

What is a random variable?

Back

A random variable is a numerical outcome of a random phenomenon, which can take on different values based on the outcome of a random event.

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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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What does standard deviation measure in a probability distribution?

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Standard deviation measures the amount of variation or dispersion of a set of values; a low standard deviation indicates that values tend to be close to the mean, while a high standard deviation indicates that values are spread out over a wider range.

Front

What is the significance of the mean in a probability distribution?

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The mean is the central value of a probability distribution, representing the average outcome of a random variable.

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What is a binomial distribution?

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

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What is the formula for calculating the expected value of a sum of random variables?

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The expected value of a sum of random variables is the sum of their expected values: E(X + Y) = E(X) + E(Y).

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What does it mean for random variables to be independent?

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Two random variables are independent if the occurrence of one does not affect the probability of the occurrence of the other.

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What is the empirical rule in statistics?

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The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations.

Front

How do you interpret a standard deviation of 1.08 in the context of a random variable?

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A standard deviation of 1.08 indicates that the values of the random variable typically vary by about 1.08 units from the mean.

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What is the relationship between the mean and standard deviation in a normal distribution?

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In a normal distribution, the mean indicates the center of the distribution, while the standard deviation indicates the spread of the data around the mean.