LGT5101 Chap.6 Continuous Variables and Normal Models
Continuous Variables and Normal Models
Probability becomes area
A continuous random variable takes infinitely many values across an interval and assigns probability through area under a density curve. Total area is one, while a single exact point has area zero. Inclusive and exclusive endpoints therefore give the same continuous interval probability.
Density height is not probability by itself; only area over a range answers an event.
Use the cdf as a left-tail engine
The cumulative distribution F(x) returns the area to the left of x. A right-tail probability is one minus that cdf, and a between-area is the upper cdf minus the lower cdf. Sketching the boundary and shading the event before using software prevents tail reversal.
Keep unrounded standard scores during calculation, then report a probability with the event and original units.
Standardise the normal model
A normal distribution is symmetric and controlled by mean and standard deviation. Mean, median and mode coincide, while standard deviation sets horizontal spread. Standardisation maps a boundary to z=(x-mu)/sigma without changing its probability area.
The standard normal cdf then handles every normal location and scale. Empirical-rule percentages provide quick plausibility checks only when the distribution is approximately bell-shaped.
Reverse the calculation for a percentile
Inverse cdf calculations receive a left-tail probability and return its boundary.
Convert wording such as exceeded on five per cent of days into a ninety-five per cent left tail before applying NORM.INV. When approximating a binomial count, check the course conditions np greater than ten and n(1-p) greater than ten, and use mean np and variance np(1-p). The smooth curve remains an approximation to integer bars.
Finally, test model fit and direction: a percentile above fifty per cent must lie above the mean, and a boundary above the mean must have left-tail probability above one half.
What this chapter covers
- 01
Continuous random variables
- 02
Probability density
- 03
Cumulative distribution
- 04
Normal parameters
- 05
Standard normal variable
- 06
Interval and tail probability
- 07
Normal approximation
- 08
Inverse cdf
Worked example · free
Delivery time between two boundaries
- 3Sketch and standardise both boundaries.
- 3Subtract the two cumulative areas.
- 3Interpret the probability under the model.
Key terms
- Density Curve
- A continuous curve whose interval area represents probability.
- Cumulative Distribution
- The probability area to the left of a stated boundary.
- Normal Distribution
- A symmetric bell-shaped distribution determined by mean and standard deviation.
- Standard Normal
- A normal variable with mean zero and standard deviation one.
- Normal Quantile
- The boundary leaving a stated cumulative probability to its left.
Continuous Variables and Normal Models FAQ
Why is the probability of one exact continuous value zero?
Probability is represented by area under the density curve. One point has no width and therefore no area, even when the density height there is positive. As a result, including or excluding the endpoints of a continuous interval does not change its probability.
Why do right-tail questions need a complement?
A normal cdf returns probability to the left of a boundary. The area to the right is what remains from the total area one, so P(X greater than x) equals one minus the cdf at x. Sketching the shaded side prevents reversal.
How do I read an inverse-normal question?
Convert the wording into a left-tail probability first. A threshold exceeded on 5% of occasions has 95% to its left, so the inverse function receives 0.95. Entering the visible 0.05 would return the opposite, lower-tail threshold.
Exam move
Sketch a bell curve for every normal question and shade the requested region before using software. Label forward questions as boundary to area and inverse questions as area to boundary. Practise left tails, right-tail complements and between-area differences. Use the empirical rule as a plausibility check, then translate the final probability or quantile back into the original unit and decision context.
Make a two-column log in which each probability statement is paired with its Excel expression and a sentence naming the shaded area. For inverse problems, rewrite upper-tail wording as a left-tail probability before using NORM.INV, then check whether the returned threshold lies on the expected side of the mean.
For binomial approximation, calculate np and n(1-p), state the course rule and carry the variance through to the correct standard deviation. Compare exact and approximate results for one small example. Finish by inspecting a histogram for symmetry, a single peak and process stability, because precise cdf output cannot compensate for an unsuitable normal model.
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