Continuous Probability Distributions

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What is a continuous probability distribution?

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A continuous probability distribution is a probability distribution that describes the likelihood of a continuous random variable taking on a range of values. Unlike discrete distributions, continuous distributions can take any value within a given range.

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What is the mean in a probability distribution?

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The mean is the average value of a set of data points, calculated by summing all values and dividing by the number of values. In a probability distribution, it represents the center of the distribution.

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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 the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

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What is the 68-95-99.7 rule?

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

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What is a z-score?

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A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is expressed in terms of standard deviations from the mean.

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How do you find the area under a normal curve?

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The area under a normal curve can be found using z-scores and standard normal distribution tables, or by using statistical software to calculate probabilities.

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What is the difference between discrete and continuous random variables?

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Discrete random variables can take on a countable number of values (e.g., number of students), while continuous random variables can take on an infinite number of values within a given range (e.g., height, weight).

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What is the cumulative distribution function (CDF)?

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The cumulative distribution function (CDF) is a function that describes the probability that a random variable takes on a value less than or equal to a specific value.

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What is the probability density function (PDF)?

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The probability density function (PDF) describes the likelihood of a continuous random variable taking on a specific value. The area under the PDF curve over an interval represents the probability of the variable falling within that interval.

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What does it mean for a distribution to be normal?

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A normal distribution is a symmetric, bell-shaped distribution characterized by its mean and standard deviation. It follows the empirical rule and is defined by the properties of the normal curve.