Normal Distribution
The normal distribution is the main approximation tool on Exam P: it turns means and variances into z-scores and helps model sums, averages, and bell-shaped uncertainty.
What normal answers
The normal distribution is the classic bell-shaped model. On Exam P, it usually matters less as a standalone object and more as the distribution you standardize, approximate with, or use to model sums and averages.
The SOA syllabus also notes that a normal table is provided, which is a signal that standardization and table lookups are part of the expected workflow rather than optional extras.
Core formulas
A normal random variable is described by a mean and a standard deviation. Standardization converts a raw value into the standard normal scale so a table can be used.
Claim severity worked example
Suppose claim severity is modeled as normal with mean 100 and standard deviation 15. The probability a claim is less than 130 is the probability that Z is less than (130 - 100) / 15 = 2.
That means the problem becomes a standard-normal left-tail lookup at 2, which is about 0.977. The key move was not the lookup itself. It was turning the raw value into a z-score first.
Normal approximation and continuity correction
Normal approximation is often used for sums or counts when the exact distribution is inconvenient. For a discrete count, the continuity correction moves the cutoff by 0.5 before standardizing.
If a binomial count has mean 80 and standard deviation 4, the approximation to the probability of at most 75 successes uses 75.5 before converting to a z-score.
How this appears on Exam P
Normal questions often appear in one of three ways: direct probability with a given mean and variance, approximation to a discrete model, or linear-combination and aggregation logic.
That makes the normal distribution one of the highest-impact Exam P pages because it connects univariate formulas to multivariate and approximation questions.
Exam traps
A common mistake is forgetting that standardization needs the standard deviation, not the variance. Another is using a normal approximation without checking whether the setup is actually reasonable for the underlying count or sum.
Candidates also lose easy points by treating the table as mysterious. The table is just the last step. The real skill is setting up the standardized value correctly.
- Use standard deviation in the denominator, not variance.
- For greater-than probabilities, check whether the table gives left-tail area.
- For discrete approximations, decide whether a continuity correction is needed before standardizing.
- For sums of independent normal variables, add means and variances, not standard deviations.
Modeling connection
In statistics and ML, the normal distribution shows up in error models, Gaussian assumptions, approximation arguments, and uncertainty summaries. On Exam P, the same intuition appears in a more elementary form through z-scores and normal approximation.
Original exam practice
4 questions built from syllabus outcomes and released-exam patterns. The prompts and answers are original, so they train the skill without copying official exam text.
Normal Z-Score and Approximation Drill
Original normal checks for standardization, table direction, continuity correction, and variance of sums.
- Question 1/Calculation
Claim severity z-score
Claim severity is modeled as normal with mean 100 and standard deviation 15. Convert a claim amount of 130 to a z-score.
Solution and grading points
Subtract the mean and divide by the standard deviation: (130 minus 100) divided by 15 equals 2.
- Subtracts the mean from the raw value.
- Divides by the standard deviation, not the variance.
- Reports z equal to 2.
- Question 2/Written Answer
Right tail from a left-tail table
A standard normal table gives left-tail area. How would you find the probability that a standard normal variable is greater than 1.4?
Solution and grading points
Look up the left-tail area at 1.4, then subtract it from 1. The greater-than probability is the right tail.
- Identifies the table as left-tail.
- Uses a complement.
- Keeps the direction of the inequality clear.
- Question 3/Calculation
Continuity correction cutoff
A discrete count is approximated by a normal random variable. For the probability of at most 75 counts, what cutoff should be used before standardizing?
Solution and grading points
Use 75.5 as the continuity-corrected cutoff because at most 75 includes all values up through 75.
- Recognizes that a continuity correction is needed.
- Moves the upper cutoff to 75.5.
- Keeps the at-most direction intact.
- Question 4/Calculation
Variance of an independent sum
Two independent normal losses have means 100 and 60, with standard deviations 15 and 8. Find the mean and standard deviation of their sum.
Solution and grading points
The mean is 160. The variance is 15 squared plus 8 squared, or 289, so the standard deviation is 17.
- Adds the two means.
- Adds variances because independence is stated.
- Takes the square root after adding variances.