Exam P Uniform Distribution Practice Problems

Try this original problem before opening the solution. Then compare the method with SOA Exam P sample question 230.

Original practice · fully worked

Original variant: variance ratio after squaring a symmetric reading

A calibration reading X is uniformly distributed from -1 to 1. A diagnostic display records Y=X². Calculate the ratio Var(Y)/Var(X).

A
0.0889
B
0.2667
C
0.3333
D
0.6000
E
1.0000

Show the answer and worked solution

Answer and calculation

For symmetric Uniform(-1,1), Var(X)=1/3. Since Y=X squared, E[Y]=1/3 and E[Y squared]=1/5, giving Var(Y)=4/45. The ratio is 4/15=0.2667, choice B.

Setup

Use symmetry to obtain the first two moments of the original reading.

\operatorname{E}[X]=0

\operatorname{E}[X^2]=\frac13

\operatorname{Var}(X)=\frac13

Model

Translate the display moments into even raw moments of X.

\operatorname{E}[Y]=\operatorname{E}[X^2]=\frac13

\operatorname{E}[Y^2]=\operatorname{E}[X^4]=\frac15

Compute

Form the display variance and divide by the reading variance.

\operatorname{Var}(Y)=\frac15-\left(\frac13\right)^2=\frac4{45}

\frac{\operatorname{Var}(Y)}{\operatorname{Var}(X)}=\frac{4/45}{1/3}=\frac4{15}=0.2666667

Answer

The variance ratio is approximately 0.2667.

\boxed{0.2667\quad\text{(B)}}

Technique orientation

What uniform distribution questions are testing

A uniform model assigns equal density across a continuous interval or equal mass across a finite set. That distinction matters: continuous probabilities are ratios of lengths, while discrete probabilities are ratios of counts. Start by writing the support and determine which version the problem uses.

Exam P combines uniform variables with conditioning, transformations, maxima and minima, and geometric regions. For one variable, the key is the overlap between the requested interval and the support. For joint uniform models, probability becomes an area or volume ratio only when the density is constant on the stated region.

Recognition audit

Signals that this is the right method

Every point in an interval has the same density, or every listed outcome has the same probability mass.
The desired probability can be represented by the length, area, or volume of a subregion of the support.
A linear transformation changes the endpoints but preserves uniformity.
A maximum, minimum, or order statistic is built from independent uniform observations.

Before calculation

A disciplined first pass

Name the requested quantity, mark its units or time point, and identify the assumption that permits the formula. Only then insert the numerical inputs. This separates model errors from arithmetic errors and makes the answer-choice comparison auditable.

Formula core

Uniform Distribution formulas to know

These identities are starting points, not substitutes for checking support, timing, rate periods, or independence assumptions.

Continuous uniform density

X\sim U(a,b)\quad\Longrightarrow\quad f_X(x)=\frac{1}{b-a},\qquad a\le x\le b

Probability is the supported interval length multiplied by 1/(b − a).

Mean and variance

E[X]=\frac{a+b}{2},\qquad \operatorname{Var}(X)=\frac{(b-a)^2}{12}

The variance depends on the width of the interval, not on its location.

Maximum of independent uniforms

P(\max(X_1,\ldots,X_n)\le x)=\left(\frac{x-a}{b-a}\right)^n,\qquad a\le x\le b

All n observations must fall below x, which is where independence enters.

Three failure modes to catch

Using a continuous length ratio for a discrete uniform model, where endpoints change the outcome count.
Measuring the requested interval without clipping it to the distribution’s support.
Assuming a joint support is rectangular when its stated boundaries form a triangle or another constrained region.

Verified practice ledger

Exam P uniform distribution problems and solutions

The list is grouped by current official syllabus area. Every link opens a step-labeled solution, verification artifact, distractor analysis, and one fully worked original variant.

General Probability

1 verified reference

SOA Exam P Sample Question #348IndependenceVerified

Univariate Random Variables

48 verified references

Multivariate Random Variables

1 verified reference

SOA Exam P Sample Question #245Conditional VarianceVerified

Continue the study loop

Keep every worked reference in one searchable manual

The complete Exam P manual includes all active sample-reference solutions, their wrong-choice analysis, and one independently written worked variant per reference. The free formula sheet is the compact companion for recall between practice sessions.

View contents and sample pagesGet the free formula sheet

Related drills

Other Exam P technique collections

Switch techniques when you can identify the current method before writing a formula and can explain each wrong-choice mechanism.

103 verified problems

Uniform Distribution practice problems