STATISTICS & PROBABILITY 11: Definition of Terms + Formulas for 1st Semester Midterm

PLEASE TAKE NOTE THIS IS ONLY DEFINITION OF TERMS AND A FEW FORMULAS FOR THE ENTIRE MIDTERM COVERAGE. For in-depth material, refer to your book or previous worksheets. Thank you and Godbless! :)

Random Variable

A function that associates a real number to each element in the sample space. it is a variable whose values are determined by chance.

Discrete Random Variable

A countable, finite random variable.

Continuous Random Variable

• A random variable that takes on values from within an interval or disjoint union of intervals.

• These often represent measured data, such as heights, weights, and temperatures.

• Additionally, this type of random variable is infinite.

Numerical Data

• Also referred to as quantitative data.

• These are quantities that can be counted or measured.

• Classified into 2 types: continuous and discrete.

Categorical Data

• Also referred to as qualitative data.

• These represent characteristics or qualities that can be grouped.

• Classified into 2 types: nominal and ordinal.

Nominal

A type of categorical where categories do not have any inherent order.

Ordinal

A type of categorical variable where categories have a meaningful order or rank.

Probability Distribution Function

A function P(X) that shows the relative probability that each outcome of an experiment will happen.

Probability Mass Function

A probability distribution function of a discrete random variable

Discrete Probability Distribution

A table of values that shows the probability of any of the outcomes of an experiment.

Probability Histogram

It is a graph of the probability distribution that displays the possible values of a discrete random variable on the horizontal axis and the probabilities of those values on the vertical axis. The probability of each value is represented by a vertical bar whose height equals its probability.

Non-negativity property

A property that states that every probability value must be greater than or equal to zero. Probabilities cannot be negative.

Norming property

A property that states that the sum of probabilities of all possible outcomes in a sample space must equal 1. This ensures that some outcome will definitely occur.

Equally likely

The term for when all the probabilities are the same.

Mean

A weighted average of the possible values that the random variables can take.

• μ = Σ [x * P(x)]

Variance

The measure of the spread of dispersion. It measures the variation of the values of a random variable from the mean.

• σ² = Σ[(x - μ)² * p(x)]

Standard Deviation

The root of the variance.

• σ = ∑√[(x − μ)² × P(x)]

Normal Distribution

• It is a distribution of a continuous random variable whose graph is a bell-shaped curve called normal curve.

• Also known as the Gaussian Distribution, in honor of the renowned German mathematician Johann Carl Friedrich Gauss

Properties of the Standard Normal Distribution

1. The graph of the normal distribution is bell-shaped and extends indefinitely in both directions.

2. Symmetrical about the y-axis.

3. The mean, median, and mode coincide at the center.

4. The distribution is unimodal.

5. Its curve is asymptotic with respect to the x-axis.

6. The total area under its curve is 1.

Z-Score

This tells how many standard deviations a value is away from the mean.

Standardizing a Normal Distribution

The process of converting a normal distribution with any mean and standard deviation into a standard normal distribution with a mean of 0 and a standard deviation of 1. This involves changing the X-scale of the variable into the Z-scale.

• z = (X – μ) / σ