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What is the mean of a hyper-geometric variable represented as?
What is the mean of a hyper-geometric variable represented as?
The variance of a hyper-geometric variable can be calculated using the formula $Var(X) = npq \cdot \frac{N - n}{N - 1}$.
The variance of a hyper-geometric variable can be calculated using the formula $Var(X) = npq \cdot \frac{N - n}{N - 1}$.
True
In a lot containing 30 items, 6 are defective. What is the probability of selecting a random sample of five items containing no defective items?
In a lot containing 30 items, 6 are defective. What is the probability of selecting a random sample of five items containing no defective items?
0.4783
The expected value of the number of red chips drawn when taking two chips out of a bowl containing six red and four blue chips is ______.
The expected value of the number of red chips drawn when taking two chips out of a bowl containing six red and four blue chips is ______.
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Match the following terms with their definitions:
Match the following terms with their definitions:
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What type of random variable is used to calculate the probability of getting the first success on a specific trial?
What type of random variable is used to calculate the probability of getting the first success on a specific trial?
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The hypergeometric distribution deals with sampling with replacement.
The hypergeometric distribution deals with sampling with replacement.
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What is the probability that the 10th person you encounter in a town with 4% teachers is a teacher?
What is the probability that the 10th person you encounter in a town with 4% teachers is a teacher?
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The mean of a geometric distribution with probability of success $p$ is _____
The mean of a geometric distribution with probability of success $p$ is _____
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Match the terms with their definitions:
Match the terms with their definitions:
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If 2% of tires produced by a company are defective, what is the probability that the first defect is found in the first 5 samples?
If 2% of tires produced by a company are defective, what is the probability that the first defect is found in the first 5 samples?
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The variance of a geometric distribution with probability $p$ is calculated as $p/(1-p)^2$.
The variance of a geometric distribution with probability $p$ is calculated as $p/(1-p)^2$.
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What would you expect the number of tires to test until the first defective one is found if 2% of tires are defective?
What would you expect the number of tires to test until the first defective one is found if 2% of tires are defective?
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What does a geometric random variable represent in probability theory?
What does a geometric random variable represent in probability theory?
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The probability of needing x trials for the first success in a geometric distribution decreases as x increases.
The probability of needing x trials for the first success in a geometric distribution decreases as x increases.
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Define a geometric random variable.
Define a geometric random variable.
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In a geometric distribution, the probability of success on each trial is denoted by ______.
In a geometric distribution, the probability of success on each trial is denoted by ______.
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If a fair coin is tossed until a head appears, what is the probability that exactly three tosses are needed?
If a fair coin is tossed until a head appears, what is the probability that exactly three tosses are needed?
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In the scenario of rolling a die until a six appears, the probability of needing at least 3 rolls is higher than needing at most 4 rolls.
In the scenario of rolling a die until a six appears, the probability of needing at least 3 rolls is higher than needing at most 4 rolls.
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What is the formula to calculate the probability that at most x trials are needed for the first success in a geometric distribution?
What is the formula to calculate the probability that at most x trials are needed for the first success in a geometric distribution?
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Match the following scenarios with their corresponding probabilities:
Match the following scenarios with their corresponding probabilities:
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Study Notes
Probability Distributions
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Geometric Random Variable: A Bernoulli trial is repeated until a success occurs. X represents the number of trials before the first success. The probability of x trials before the first success is given by (1-p)^(x-1) * p, where p is the probability of success on a single trial.
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Probability Density Function (PDF) for Geometric Random Variable: A discrete random variable, X, is geometric if its PDF is:
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P(X = x) = (1-p)^(x-1) * p, for x = 1, 2, ...
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0, otherwise
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Mean and Variance of Geometric Distribution:
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Mean (Expected Value): E(X) = 1/p
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Variance: Var(X) = (1-p)/p²
Hypergeometric Distribution
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Key Difference: Sampling without replacement from a finite population. This is in contrast to the binomial distribution, which deals with sampling with replacement.
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Probability Density Function (PDF) for Hypergeometric Distribution:
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f(x) = [ (N₁ choose x) * (N - N₁ choose n - x) ] / (N choose n), for x, n—x >= 0 Where:
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N = Total population size
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N₁ = Number of items possessing a particular characteristic
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n = Sample size
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Constraints: N ≥ N₁, N ≥ n, and n − x ≤ N − N₁ are crucial for the formula to apply.
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Description
Test your knowledge on probability distributions, including geometric and hypergeometric distributions. This quiz will cover key concepts such as probability density functions, means, variances, and the differences between sampling with and without replacement.
