Statistics: Expected Value, Variance, and
Probability Distributions Complete 2026–
2027 Exam Preparation Guide: Updated
Question Bank, Explained Solutions, Concept
Review & Effective Study Strategies
Description
The Complete 2026–2027 Exam Preparation Guide is a
structured and reliable resource developed to support
students in achieving better academic outcomes through
focused practice, clear explanations, and organized
revision.
This guide is designed to bridge the gap between learning
concepts and applying them in exams. By combining a
wide range of practice questions with clearly explained
solutions, it helps students understand not only the correct
answers but also the methods required to solve similar
problems independently.
E(aX+b) - ANSWER Expected value transformation formula: E(aX+b)=aE(X)+b. Use when
finding mean/expected profit, revenue, cost, etc.
,Var(aX+b) - ANSWER Variance transformation formula: Var(aX+b)=a²Var(X). Adding or
subtracting a constant does not affect variance.
Fixed cost and variance - ANSWER A fixed cost like -80 does not affect variance because it
does not change day-to-day spread.
Profit formula - ANSWER If X is quantity sold and price is p with fixed cost C, profit = pX - C.
Variance of profit - ANSWER If profit = pX - C, then Var(profit)=p²Var(X).
Expected profit - ANSWER If profit = pX - C, then E(profit)=pE(X)-C.
Discrete expected value - ANSWER E(X)=Σ xP(X=x). Multiply each possible value by its
probability and add.
Discrete variance formula - ANSWER Var(X)=E[(X-μ)²], where μ=E(X).
Variance shortcut - ANSWER Var(X)=E(X²)-[E(X)]².
E(X²) - ANSWER E(X²)=Σ x²P(X=x). Square each x first, multiply by its probability, then add.
Continuous expected value - ANSWER E(X)=∫ x f(x) dx over the support of X.
Probability from pdf - ANSWER P(a≤X≤b)=∫ from a to b f(x) dx.
, Continuous exact probability - ANSWER For a continuous random variable, P(X=exact
number)=0.
PDF - ANSWER A probability density function. Probabilities are areas under the curve, not the
height itself.
Support - ANSWER The values where a random variable can actually exist / where the pdf or
pmf is nonzero.
Uniform distribution - ANSWER X~Uniform(a,b) means X is evenly spread between a and b.
Uniform pdf - ANSWER f(x)=1/(b-a) for a≤x≤b, and 0 otherwise.
Uniform mean - ANSWER E(X)=(a+b)/2.
Uniform quantile - ANSWER x=a+q(b-a), where q is the percent below x.
Uniform "80% above" question - ANSWER If P(X≥P)=0.80, then 20% is below P, so use q=0.20
in x=a+q(b-a).
Complement rule - ANSWER P(Aᶜ)=1-P(A).
At least one shortcut - ANSWER P(at least one)=1-P(none).
Union - ANSWER A∪B means A or B or both happen.
Probability Distributions Complete 2026–
2027 Exam Preparation Guide: Updated
Question Bank, Explained Solutions, Concept
Review & Effective Study Strategies
Description
The Complete 2026–2027 Exam Preparation Guide is a
structured and reliable resource developed to support
students in achieving better academic outcomes through
focused practice, clear explanations, and organized
revision.
This guide is designed to bridge the gap between learning
concepts and applying them in exams. By combining a
wide range of practice questions with clearly explained
solutions, it helps students understand not only the correct
answers but also the methods required to solve similar
problems independently.
E(aX+b) - ANSWER Expected value transformation formula: E(aX+b)=aE(X)+b. Use when
finding mean/expected profit, revenue, cost, etc.
,Var(aX+b) - ANSWER Variance transformation formula: Var(aX+b)=a²Var(X). Adding or
subtracting a constant does not affect variance.
Fixed cost and variance - ANSWER A fixed cost like -80 does not affect variance because it
does not change day-to-day spread.
Profit formula - ANSWER If X is quantity sold and price is p with fixed cost C, profit = pX - C.
Variance of profit - ANSWER If profit = pX - C, then Var(profit)=p²Var(X).
Expected profit - ANSWER If profit = pX - C, then E(profit)=pE(X)-C.
Discrete expected value - ANSWER E(X)=Σ xP(X=x). Multiply each possible value by its
probability and add.
Discrete variance formula - ANSWER Var(X)=E[(X-μ)²], where μ=E(X).
Variance shortcut - ANSWER Var(X)=E(X²)-[E(X)]².
E(X²) - ANSWER E(X²)=Σ x²P(X=x). Square each x first, multiply by its probability, then add.
Continuous expected value - ANSWER E(X)=∫ x f(x) dx over the support of X.
Probability from pdf - ANSWER P(a≤X≤b)=∫ from a to b f(x) dx.
, Continuous exact probability - ANSWER For a continuous random variable, P(X=exact
number)=0.
PDF - ANSWER A probability density function. Probabilities are areas under the curve, not the
height itself.
Support - ANSWER The values where a random variable can actually exist / where the pdf or
pmf is nonzero.
Uniform distribution - ANSWER X~Uniform(a,b) means X is evenly spread between a and b.
Uniform pdf - ANSWER f(x)=1/(b-a) for a≤x≤b, and 0 otherwise.
Uniform mean - ANSWER E(X)=(a+b)/2.
Uniform quantile - ANSWER x=a+q(b-a), where q is the percent below x.
Uniform "80% above" question - ANSWER If P(X≥P)=0.80, then 20% is below P, so use q=0.20
in x=a+q(b-a).
Complement rule - ANSWER P(Aᶜ)=1-P(A).
At least one shortcut - ANSWER P(at least one)=1-P(none).
Union - ANSWER A∪B means A or B or both happen.
