Probability For Continuous Variables

What’s the key difference between discrete and continuous variables?

Discrete variables have separate categories; continuous variables can take on infinite values between integers.

Why do we assess probability in ranges for continuous variables?

Because the probability of any single exact value is effectively zero.

What kind of function do continuous variables use to describe probability?

A probability density function (PDF).

How do PDFs differ from PMFs (used for discrete variables)?

PDFs are smooth curves; PMFs are bar-like functions.

What’s the total area under a PDF curve?

1 (representing 100% of the probability).

Why are PDFs used to represent populations?

Because we often don't know all population values, we assume their distribution follows a theoretical model (like the normal distribution).

What values must we often estimate in PDFs?

Population mean (μ) and standard deviation (σ).

What are key properties of the normal distribution?

• Symmetrical and mesokurtic (𝑔₁ = 0, 𝑔₂ = 0)

• Mean = Median = Mode

• Unimodal

• Inflection points at μ ± σ

What are the empirical rule percentages for a normal curve?

• ~68% within ±1σ

• ~95% within ±2σ

What does notation X∼N(μ,σ²) mean?

Variable X follows a normal distribution with mean μ and variance σ².

What is a z-score?

A standardized score representing the number of standard deviations a value is from the mean.

What is the formula to convert a raw score to a z-score?

z=x−μ​

______

σ

What is the formula to convert a z-score back to a raw score?

x=zσ+μ

What’s the “z-distribution”?

A standard normal distribution with μ = 0 and σ = 1.

What does a z-table provide?

Proportions of the area under the normal curve for specific z-values (i.e., probabilities).

Why do we use z-tables?

To find probabilities associated with any score in a normal distribution, even if it’s not exactly 1 or 2 SDs away from the mean.

What is P(X≥31)?

Convert 31 to a z-score and find area to the right under the curve.

What is P(X≤24.5)?

Convert 24.5 to z and look up the left-tail area in the z-table.

What is P(30≤X≤40)?

Convert both 30 and 40 to z-scores and subtract the smaller area from the larger.

What is P(X≤24.5 or X≥40)?

Add the left-tail and right-tail probabilities beyond those z-scores.