Stats/Module3-1

This set contains 100 flashcards covering key vocabulary and concepts from the lecture on Uniform and Normal Distributions.

Density Curve

A model of the distribution of a continuous random variable.

Uniform Distribution

A distribution where a random variable is equally likely to fall anywhere within a specified interval.

P(X ∼ Unif(a, b))

Notation for a variable X that is uniformly distributed between a and b.

Nonnegativity

The property that the value of a density function f(x) must be greater than or equal to zero for all x.

Total Area = 1

A key property of density curves indicating that the total probability sums to 1.

Probability = Area

The probability of an event is equal to the area under the density curve for that event.

Standard Normal Distribution

A normal distribution where the mean (μ) is 0 and standard deviation (σ) is 1.

Empirical Rule (68–95–99.7)

Rule stating that for a normal distribution, approximately 68% of values fall within 1 standard deviation, 95% within 2, and 99.7% within 3.

Standardization

The process of converting scores from different distributions to a common scale, typically the standard normal distribution.

Z-score

The number of standard deviations a data point is from the mean, calculated as Z = (X - μ) / σ.

Mean of Uniform Distribution

Calculated as μ = (a + b) / 2.

Variance of Uniform Distribution

Calculated as σ² = (b - a)² / 12.

Symmetric Distribution

A distribution where the left half is a mirror image of the right half.

Histogram

A graphical representation of the distribution of numerical data, using bars to show the frequency of values.

Support of a Distribution

The range of values where the probability density function is greater than zero.

Continuous Random Variable

A variable that can take any value within a given range.

Left-Tail Probability

The probability of a random variable being less than a certain value.

Right-Tail Probability

The probability of a random variable being greater than a certain value.

Interval Probability

The probability that a random variable falls within a specific range.

Percentile

A value below which a given percentage of observations in a group falls.

Chebyshev's Theorem

A statistical rule that states that no more than 1/k² of the distribution's values can be more than k standard deviations away from the mean.

Empirical Observation

The process of validating a theoretical concept through experimentation or direct observation.

Normal Distribution

A continuous probability distribution characterized by its bell-shaped curve, defined by mean and standard deviation.

Shape of Normal Distribution

Symmetric around the mean, the peaks at the mean, and tails extending indefinitely.

Tail of Distribution

The ends of a probability distribution, where extreme values are found.

Total Probability Theorem

A way to compute the total probability of an event by considering all possible scenarios.

Area Under the Curve (AUC)

The area within a density function, used to determine probabilities.

Probability Density Function (PDF)

A function that describes the likelihood of a continuous random variable taking on specific values.

Random Sampling

The process of selecting a subset of individuals from a population in such a way that every individual has an equal chance of being chosen.

Sampling Distribution

The probability distribution of a statistic obtained by sampling from a population.

Standard Error

The standard deviation of a sampling distribution, used to measure the accuracy of a sample estimate.

Outlier

A value that lies outside the general range of a set of values.

Confidence Interval

A range of values that is likely to contain the parameter being estimated.

T-distribution

A type of probability distribution used when estimating population parameters when the sample size is small.

Central Limit Theorem

States that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases.

Bias in Sampling

A systematic error introduced into sampling or testing by selecting or encouraging one outcome over others.

Uniform Probability Density Function

f(x) = 1/(b - a) for a ≤ x ≤ b, and 0 otherwise.

Normal Probability Density Function

f(x) = (1/(σ√(2π)))e^(-(x - μ)²/(2σ²)).

Cumulative Distribution Function (CDF)

A function that shows the probability that a random variable is less than or equal to a certain value.

Bivariate Distribution

A probability distribution involving two random variables.

Joint Probability

The probability of two events happening at the same time.

Marginal Probability

The probability of an event without consideration of any other events.

Conditional Probability

The probability of an event given that another event has occurred.

Data Shift

The adjustment of data to reflect a different mean or variance.

Computational Probability

The calculation of probabilities using mathematical formulas and rules.

Statistical Significance

A determination that an observed effect in data is unlikely to be due to chance alone.

Fuzzy Sets

Sets that allow for partial membership, as opposed to classical binary sets.

Robustness in Statistics

The quality of a statistical method to perform well under various conditions.

Monte Carlo Simulation

A computational algorithm that relies on repeated random sampling to obtain numerical results.

Bayesian Statistics

A method of statistics in which probabilities are interpreted as measures of belief.

Frequentist Statistics

A perspective in statistics that emphasizes the frequency of events occurring based on sampled data.

Qualitative Data

Non-numeric data that describes qualities or characteristics.

Quantitative Data

Numeric data that can be measured and expressed mathematically.

Descriptive Statistics

Statistical methods that summarize and describe the characteristics of a data set.

Inferential Statistics

Techniques that allow conclusions to extend beyond an immediate data set.

Sampling Bias

A bias that occurs when the sample is not representative of the population.

Observational Study

A study where the researcher observes and records data without manipulation.

Controlled Experiment

An experiment where all variables are controlled except for the ones being tested.

Statistical Inference

The process of using data analysis to deduce properties of the population.

Point Estimator

A statistic used to estimate the value of a population parameter.

Interval Estimator

A range of values used to estimate a population parameter.

Null Hypothesis

A statement of no effect, no difference or no relationship between variables.

Alternative Hypothesis

The hypothesis that there is an effect, difference or relationship.

p-value

The probability of observing the test results under the null hypothesis.

Type I Error

Rejecting the null hypothesis when it is actually true.

Type II Error

Not rejecting the null hypothesis when it is false.

Statistical Power

The probability that a test will correctly reject a false null hypothesis.

Effect Size

A measure of the strength of the relationship between two variables.

Variance Reduction

A technique aimed at decreasing the variability of an estimator.

Covariance

A measure of the degree to which two variables change together.

Correlation Coefficient

A statistical measure that describes the strength and direction of a relationship between two variables.

Regression Analysis

A statistical process for estimating the relationships among variables.

Time Series Analysis

A statistical technique that deals with time series data, or trend analysis.

Data Mining

The practice of examining large pre-existing databases to generate new information.

Machine Learning

A field of computer science that uses statistical techniques to give computers the ability to learn from data.

Data Visualization

The graphic representation of information and data.

Bayesian Network

A graphical model that represents a set of variables and their conditional dependencies.