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What is the cumulative distribution function (CDF) of the geometric distribution used for? | The probability of achieving up to a certain number of trials to achieve the first success |
What are Bernoulli Trials in the context of the geometric distribution? | Independent trials with two possible outcomes and a constant probability of success p |
How does the pmf of the geometric distribution calculate the probability of needing k trials to achieve the first success? | P(X=k)=(1−p)k−1⋅p |
What does the hypergeometric distribution model? | The number of successes in a fixed number of draws without replacement from a finite population |
What type of probability distribution is the hypergeometric distribution? | Discrete probability distribution |
How does the pmf of the hypergeometric distribution calculate the probability of obtaining exactly k successes in a sample? | P(X=k)=(K¦k)((N-K)¦(n-k))/((N¦n) ) |
What distinguishes the hypergeometric distribution from the binomial distribution? | Accounting for sampling without replacement |
What is the population size parameter in the hypergeometric distribution? | N |
In which situations is the hypergeometric distribution commonly used? | Modeling the probability of obtaining a specific number of successes in a sample drawn without replacement from a finite population |
What is one challenge related to the hypergeometric distribution? | Finite Population Assumption |
What are examples of scenarios where the hypergeometric distribution can be applied? | Sampling without replacement from a deck of cards, Quality control scenarios |
What is the sample size parameter in the hypergeometric distribution? | n |
What does the hypergeometric distribution assume about the samples drawn from the population? | They are drawn without replacement |
What is the definition of the marginal distribution? | Obtained by summing (or integrating) the joint distribution over all possible values of the other variable(s) |
What type of probability distribution is the marginal distribution? | Can be discrete or continuous |
What is one challenge related to the marginal distribution? | Loss of Information |
How is the marginal probability mass function (PMF) obtained for discrete random variables? | By summing probabilities over the values of the other variables |
In which situations are marginal distributions commonly used? | Analyzing individual variables in a multivariate setting |
What does the joint distribution describe? | Probabilities associated with all possible combinations of values that two random variables can take simultaneously |
What distinguishes the chi-square distribution from other distributions? | It is skewed to the right and has no negative values |
What concept is important in using the chi-square distribution for hypothesis testing and confidence interval estimation? | Degrees of Freedom (ν) |
What challenge is associated with the chi-square distribution? | Sensitivity to Sample Size |
How is the joint probability mass function (PMF) represented for continuous random variables? | As a joint probability density function (PDF) |
What is the uniform distribution characterized by? | A constant probability density function (pdf) over the range of possible values |
How does the cumulative distribution function (CDF) behave in a uniform distribution? | Increases linearly within the range from 0 to 1 |
In what type of scenarios is the uniform distribution commonly used? | Simulations, random number generation, and modeling situations where all outcomes are equally likely |
What is one challenge associated with the uniform distribution? | Limited Applicability |
What is an example of a continuous uniform distribution? | Selecting a random number from a specified interval [a, b] with equal probability for any value within that interval |
What concept is central to the uniform distribution's probability density function (pdf)? | Equal probability for all outcomes within the specified range |
What distinguishes the uniform distribution from other distributions in terms of probability density function (pdf)? | Constant pdf over the range of possible values |
What is one usage of the uniform distribution in statistical experiments? | Modeling scenarios where there is no reason to favor one outcome over another within the specified range |
What type of probability distribution is the uniform distribution? | Continuous and can be discrete |
What does the uniform distribution model within a given range? | All outcomes are equally likely |
What are the two different types of data in statistics? | Discrete data and continuous data |
Define discrete data. | Discrete data involves variables that have specific values that cannot have values between them. |
Give an example of a discrete variable. | The number of times someone visits their neighbors during the week. |
Why is the number of times someone visits their neighbors during the week considered a discrete variable? | Because it has specific values and cannot have values between them. |
Are categorical variables considered discrete or continuous variables? | Discrete variables |
What is continuous data? | Variables in which there could be an infinite number of values between a set range of possible values |
Give an example of a continuous variable. | Final exam grades on a scale of 0 to 100 |
Why are final exam grades considered a continuous variable? | Because there could be an infinite number of values between them |
What are things measured in time considered as? | Continuous variables |
Can someone express their age in a continuous manner? | Yes |
What is a symmetrical distribution? | When the pattern or trend of frequencies on the left and right side of the distribution are the same. |
Define a skewed distribution. | When the scores pile or stack up on one side and are spread out on the other. |
What are the two types of skewed distributions? | Positive skew and negative skew. |
What is a positive skew? | When the scores pile up on the lower end of the values with fewer scores at the high end. |
What is a negative skew? | When the scores pile up on the higher end of the values with fewer scores at the low end. |
What types of distributions are specifically used for discrete data? | Discrete data distributions such as the binomial distribution. |
What is the Chi-Square distribution used for? | Statistical tests such as the chi-square test for independence and goodness-of-fit tests. |
What distribution is used to model the number of events occurring in a fixed interval of time with a known average rate? | Poisson distribution |
Which distribution is symmetric and bounded between 0 and 1, often used for modeling growth or response curves in logistic regression? | Logistic distribution |
What distribution is used to model variables that are positively skewed, such as income or asset prices? | Log-normal distribution |
What is the joint distribution of two or more random variables? | Describes the probabilities of all possible combinations of values that these variables can take simultaneously. |
What distribution is used to model the time until k events occur in a Poisson process with a constant rate? | Gamma distribution |
What distribution is a generalization of the normal distribution and allows for varying degrees of tail weight and skewness? | Exponential power distribution |
What distribution is used to model the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials? | Geometric distribution |
What distribution is commonly used in analysis of variance (ANOVA) and regression analysis? | F-distribution |
What distribution is often used to model the time until failure in reliability engineering and survival analysis? | Weibull distribution |
What is the normal distribution also known as? | Gaussian distribution |
What are the two parameters that describe the normal distribution? | Mean (μ) and standard deviation (σ) |
What is the Central Limit Theorem related to the normal distribution? | The sampling distribution of the sample mean approaches a normal distribution as the sample size increases |
What is the standard normal distribution? | A special case of the normal distribution where the mean is 0 and the standard deviation is 1 |
What is one of the challenges related to the normal distribution in statistical analyses? | Assumption of Normality |
What are examples of data that often follow a normal distribution? | Heights and weights of individuals in a population, test scores such as SAT scores |
What does the binomial distribution represent? | The frequency of data with two outcomes |
What are the two possible outcomes in a binomial distribution? | Pass and fail |
What does the term "Bernoulli trials" refer to in the context of the binomial distribution? | Trials with two possible outcomes and a constant probability of success p |
How many parameters characterize the binomial distribution? | Two parameters |
What is the probability mass function (pmf) of the binomial distribution used for? | Giving the probability of observing a specific number of successes in n trials |
In which fields is the binomial distribution commonly used? | Quality control and finance |
What assumption does the binomial distribution make about the outcomes in each trial? | They are independent and identically distributed (i.i.d.) |
What is one challenge related to the binomial distribution in statistical analyses? | Independence Assumption |
What are examples of data that often follow a binomial distribution? | Flipping a fair coin n times and counting the number of heads, Testing the success rate of a new drug |
What is the cumulative distribution function (CDF) of the binomial distribution used for? | The probability of observing up to a certain number of successes in n trials |
What does the geometric distribution model? | The number of trials needed to achieve the first success in a sequence of independent trials |
What type of probability distribution is the geometric distribution? | Discrete probability distribution |
What does it mean for the geometric distribution to be memoryless? | The probability of achieving success in the next trial remains the same regardless of previous outcomes |
What is the probability mass function (pmf) of the geometric distribution used for? | Giving the probability of needing exactly k trials to achieve the first success |
In which situations is the geometric distribution commonly used? | Modeling the probability of achieving success for the first time after a series of independent trials |
What is one challenge related to the geometric distribution? | Infinite Expected Value |
What are examples of scenarios where the geometric distribution can be applied? | The number of attempts needed to score a goal in a soccer game, Tossing a fair coin until it lands heads |
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