Methods - Continuous Random Variables and Normal Distributions - Fundamentals

9 concise fundamentals covering density functions, continuous probabilities and normal calculations. Basic normal definitions remain in the existing Sampling and Confidence Intervals fundamentals set.

Density | What does a probability density function represent?

Probability is area under f(x), not the curve's height at one point. A valid density is non-negative over its domain and its total area is 1.

Continuous probability | Why is P(X = a) zero for a continuous variable?

A single point has no width, so its area is zero. Therefore P(X < a) = P(X ≤ a). Probabilities are assigned to intervals.

Continuous probability | How do you calculate interval and cumulative probabilities?

P(a ≤ X ≤ b) = ∫ₐᵇf(x) dx. The cumulative distribution F(x) = P(X ≤ x) gives area to the left; interval probability is F(b) − F(a).

Continuous averages | How do you find a continuous mean and variance?

E(X) = ∫xf(x) dx over the distribution's domain. E(X²) = ∫x²f(x) dx, then Var(X) = E(X²) − [E(X)]².

Percentiles | What are the median and a percentile?

The median has half the probability below it. A pth percentile has p% below it. Find x from F(x) = p/100; for the median use 0.5.

Normal notation | What does X ~ N(μ, σ²) mean?

X is normally distributed with mean μ and variance σ². The second entry is variance, not standard deviation. The standard normal has mean 0 and variance 1.

Normal probability | How do you find areas under a normal curve?

Use the correct mean and standard deviation, or convert bounds to z = (x − μ)/σ. A left-tail probability is a cumulative area; a right tail is 1 minus the left-tail area. A middle interval is the difference of two left-tail areas.

Inverse normal | How do you find a value from a given probability?

Convert the statement to area on the left, then use inverse normal. If working with a standard-normal value z, convert back using x = μ + zσ. For a top 10% cutoff, the left area is 0.90.

Normal spread | What does the 68-95-99.7 rule tell you?

Approximately 68% lies within μ ± σ, 95% within μ ± 2σ and 99.7% within μ ± 3σ. These are approximate checks; use the calculator for precise probabilities.