Probability Distributions and Sampling Methods

Vocabulary practice flashcards covering discrete and continuous probability distributions, sampling techniques, the Central Limit Theorem, and continuity corrections based on lecture notes.

Hypergeometric Probability Distribution

A probability distribution used for sampling without replacement from a fairly small population when the sample size is large relative to the population.

5% Rule for Hypergeometric Distribution

The guideline stating that the hypergeometric distribution should be used when the sample size is greater than $5\%$ of the total population ($n > 0.05N$).

Discrete Data

Data that is not infinitely divisible, typically consisting of distinct counts or whole units, such as people.

Continuous Data

Data that possesses infinite divisibility and can take on any value within an interval, such as time, distance, pounds, psi, miles, or centimeters.

Combinations Formula ($_nC_x$)

The mathematical formula ${_nC_x} = \frac{n!}{x!\,(n - x)!}$, used to calculate the number of ways to select $x$ items from a set of $n$ items without regard to order.

Poisson Distribution

A discrete probability distribution used to model the probability of a specific number of occurrences of an event over a specified interval of time, distance, or space.

Lambda ($\lambda$)

The mean number of occurrences of an event per given interval of time, distance, or space in a Poisson distribution, where $\lambda = n \cdot \pi$.

Poisson Probability Formula

The mathematical formula for finding the probability of $x$ occurrences: $P(x) = \frac{\lambda^x e^{-\lambda}}{x!}$, where $e \approx 2.71828$.

Poisson Distribution Skewness

A distributional property wherein the curve is always positively skewed (skewed to the right), becoming more symmetrical as the mean increases.

Poisson Mean and Variance Equality

The property of the Poisson distribution stating that the mean and the variance are equal to each other ($\text{Mean} = \text{Variance} = \lambda$).

Poisson Approximation to the Binomial Conditions

The criteria under which the Poisson distribution approximates the binomial: the number of trials must be greater than or equal to 20 ($n \ge 20$) and the probability of success must be less than or equal to $0.05$ ($\pi \le 0.05$).

Exponential Distribution

A continuous probability distribution used to model the time or interval between events, such as waiting for the next customer to arrive.

Exponential Probability Formula

The probability density function for the exponential distribution defined as $P(x) = \lambda e^{-\lambda x}$, where events are discrete but time is continuous.

Normal Distribution

A continuous probability distribution characterized by perfect symmetry around the mean, with a total area under the curve equal to $1.0$.

Normal Distribution Probability Density Function

The mathematical function defined as $f(x) = \frac{1}{\sigma \sqrt{2\pi}} \cdot e^{\left(-\frac{1}{2}\right)\left(\frac{x - \mu}{\sigma}\right)^2}$.

Standardized Normal Score ($z$)

The standardized measure of distance from the mean calculated as $z = \frac{x - \bar{x}}{sd}$, indicating the number of standard deviations a value $x$ lies from the mean.

Randomness

A condition where outcomes occur purely as a matter of chance, such as rolling dice or flipping coins.

Simple Random Sampling

A structured, non-biased sampling process where every individual in the population has an equal chance of being selected.

Convenience Sampling

A non-random sampling method where samples are selected based on easy accessibility rather than chance.

Systematic Sampling

A sampling method where a specific systematic method or interval is followed, such as selecting every 10th item or individual.

Stratified Sampling

A sampling technique commonly used in political surveys where the population is divided into distinct subgroups (e.g., political parties, racial groups, religious groups) to ensure every group is well represented.

Cluster Sampling

A sampling method where specific groups or clusters are chosen randomly first, and then samples are drawn from within those chosen clusters (e.g., choosing 10 Ohio counties).

Central Limit Theorem

The principle stating that as repeated samples of a given size are taken, the distribution of the sample means will be approximately normal, even if the underlying population from which they are drawn is not normal.

Continuity Factor (Continuity Correction)

An adjustment of $0.5$ applied when using a continuous normal distribution to approximate discrete probability distributions.

Continuity Correction for 'At Least $x$'

The rule specifying that when calculating the probability of 'at least $x$', the discrete value is adjusted downward to $x - 0.5$.

Continuity Correction for 'More Than $x$'

The rule specifying that when calculating the probability of 'more than $x$', the discrete value is adjusted upward to $x + 0.5$.

Binomial Variance Formula

The formula used to compute the variance of a binomial distribution: $\text{Var} = n \cdot \pi \cdot (1 - \pi)$, with standard deviation $sd = \sqrt{\text{Var}}$.

Uniform Probability Distribution

A continuous probability distribution where all outcomes across a given interval from $a$ to $b$ have equal likelihood, characterized by a constant height of $f(x) = \frac{1}{b - a}$ and a total area of $1$.