5.7: Vocabulary (Chapter 5)
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Chapter 5 Vocabulary
This chapter introduced several key terms and definitions related to probability distributions, random variables, and the normal distribution. Review the terms below and be prepared to use them in context when solving problems or justifying calculations.
- Random Variable
- A variable that takes numerical values determined by the outcome of a random process. Each value is associated with a probability.
- Discrete Random Variable
- A random variable with a finite or countable set of possible outcomes. Example: number of heads in 3 coin flips.
- Continuous Random Variable
- A random variable with an infinite number of possible values within a given interval. Example: time to complete a quiz.
- Probability Distribution
- A table, graph, or formula that assigns probabilities to each possible value of a random variable.
- Bernoulli Trial
- An experiment with only two possible outcomes: success or failure.
- Bernoulli Distribution
- A discrete probability distribution for a single Bernoulli trial. Only two values possible: 0 and 1.
- Binomial Distribution
- A probability distribution that models the number of successes in \( n \) independent Bernoulli trials with probability \( p \).
- Binomial Coefficient
- The number of ways to choose \( x \) successes out of \( n \) trials: \( \binom{n}{x} \).
- Probability Mass Function (PMF)
- A function that gives the probability that a discrete random variable is exactly equal to some value.
- Expected Value
- The theoretical mean of a random variable, computed as \( E(X) = \sum x \cdot P(x) \).
- Variance
- The average of the squared differences from the mean: \( \sigma^2 = \sum (x - \mu)^2 P(x) \).
- Standard Deviation
- The square root of the variance. It measures the typical distance between the data values and the mean.
- Net Gain
- The outcome of a situation after subtracting the initial cost or investment. Often used in lottery and game models.
- Normal Distribution
- A continuous, symmetric, bell-shaped distribution commonly found in natural and social processes. Defined by its mean and standard deviation.
- Density Curve
- A curve that represents the probability distribution of a continuous random variable. The area under the curve equals 1.
- Symmetric Distribution
- A distribution where the left and right halves are mirror images of each other.
- Standard Normal Distribution
- A normal distribution with mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \).
- Z-score
- A standardized value that indicates how many standard deviations a data point is from the mean: \( z = \frac{x - \mu}{\sigma} \).
- Empirical Rule (68–95–99.7 Rule)
- In a normal distribution: ~68% of data fall within 1 standard deviation, ~95% within 2, and ~99.7% within 3.
- Percentile
- The percentage of data values below a given value. Example: 90th percentile means 90% scored below that point.
- Z-table (Standard Normal Table)
- A table showing the cumulative probability up to a given z-score in the standard normal distribution.
- Area under the curve
- In a normal distribution, the area under the curve corresponds to probability. The total area is 1.
- Left Tail
- The area under the normal curve to the left of a z-score. Represents \( P(Z < z) \).
- Right Tail
- The area to the right of a z-score: \( P(Z > z) = 1 - P(Z < z) \).
- Standardization
- The process of converting raw data into z-scores so values can be compared across distributions with different scales.
- Between Probability
- The probability that a random variable falls between two values. Calculated as the difference between cumulative areas.