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- Difficulty
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- Time
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Worksheet12th GradeLevel 3 · Challenge
Free 12th Grade Continuous Random Variables and Probability Density Functions Worksheet
A senior math worksheet on continuous random variables: checking and completing probability density functions, finding probabilities by integration, calculating the mean, variance and median, moving between a density function and its cumulative distribution function, and applying the uniform and normal distributions. Finishes with an extended problem on the lifetime of a component and its warranty. Full working is given for every answer.
Instant download · No sign-up · Free for personal, classroom and homeschool use
- Grade
- 12th Grade
- Subject
- Math
- Topic
- Statistics
- Difficulty
- Level 3 · Challenge
- Estimated time
- 50 minutes
- Curriculum
- Texas grade levels (TEKS)
- Questions
- 17
- Answers
- Included, with working
- Format
- PDF (A4), print or complete online
Students will practice
- checking that a function is a valid probability density function and finding an unknown constant
- finding probabilities, the mean, variance and median of a continuous random variable by integration
- moving between a probability density function and its cumulative distribution function
- applying the uniform and normal distributions to practical problems
Curriculum: Texas grade levels (TEKS). We show specific outcome codes only where they have been verified against the official curriculum document.
What's next?
Completed: Continuous Random Variables and Probability Density Functions
More resources for this topic are coming soon.
How to use this worksheet
- Print the PDF or complete it on screen. Write the student's name and date at the top.
- Read the instructions for each part together before starting. Encourage working to be shown, not just answers.
- Mark with the answer sheet. Compare the method, not only the final answer, and give credit for correct working.
- Record the score, celebrate what went well, then follow the "What's next" suggestion.
Common questions
Who is this worksheet for?
12th Grade students (typically ages 17–18) working on statistics. It is pitched at level 3 · challenge.
Are the answers included?
Yes. The PDF includes a full answer sheet with the working for every question. You can also view the answers online.
How long does it take?
About 50 minutes for 17 questions. Short, regular sessions work best: two or three a week beats one long one.
Do I need to sign up to download?
No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.
What should we do next?
Explore more 12th Grade mathematics resources.
Preview
Success TutoringWorksheets12th Grade · Mathematics · StatisticsContinuous Random Variables and Probability Density Functions
Name
Date
Score /
Show your work. Give exact answers where possible, otherwise round as instructed.
Part A: Probability density functions
A continuous random variable can take any value in an interval (a time, a mass, a length). It is described by a probability density function (pdf) f(x), with f(x) ≥ 0 everywhere and total area 1: ∫ f(x) dx = 1 over all x.
Probabilities are areas: P(a ≤ X ≤ b) = ∫ from a to b of f(x) dx. Because a single point has no area, P(X = a) = 0, and ≤ and < give the same probability.
Worked example. f(x) = kx for 0 ≤ x ≤ 2. ∫ from 0 to 2 of kx dx = 2k = 1, so k = . Then P(X < 1) = ∫ from 0 to 1 of dx = [frac 1/4].
1.X has probability density function f(x) = kx2 for 0 ≤ x ≤ 3 (and 0 elsewhere). Find k. (answer space)
2.For the same X, with f(x) = on 0 ≤ x ≤ 3, find P(X ≤ 2). (answer space)
3.For a continuous random variable X, P(X = 1.5) = 0. ☐ True ☐ False
4.Which function is a valid probability density function on 0 ≤ x ≤ 1 (and 0 elsewhere)?
- Af(x) = 2x
- Bf(x) = x
- Cf(x) = 3x2 − 1
- Df(x) = 1.5
5.X has density f(x) = (4 − x) for 0 ≤ x ≤ 4. Find P(1 ≤ X ≤ 3).
Part B: Mean, variance and median
Mean: E(X) = μ = ∫ x f(x) dx. Variance: Var(X) = E(X2) − μ2, where E(X2) = ∫ x2 f(x) dx. Standard deviation: σ = √Var(X).
The median m splits the area in half: ∫ from the lowest value to m of f(x) dx = .
Worked example. f(x) = 2x on 0 ≤ x ≤ 1. E(X) = ∫ 2x2 dx = ; E(X2) = ∫ 2x3 dx = ; Var(X) = − = . Median: m2 = , so m = √0.5 ≈ 0.707.
6.For f(x) = on 0 ≤ x ≤ 3, find E(X).
7.For the same distribution, find Var(X).
8.For the same distribution, find the median, correct to 2 decimal places.
9.For f(x) = (4 − x) on 0 ≤ x ≤ 4, E(X) = (answer space).
Part C: Cumulative distribution functions
The cumulative distribution function (cdf) is F(x) = P(X ≤ x), the area under f to the left of x. It rises from 0 to 1, and f(x) = F′(x).
With a cdf, probabilities need no integration: P(a < X ≤ b) = F(b) − F(a), and the median solves F(m) = .
Worked example. f(x) = 2x on 0 ≤ x ≤ 1 has F(x) = x2 for 0 ≤ x ≤ 1. So P(0.5 < X < 0.8) = 0.64 − 0.25 = 0.39.
10.For f(x) = on 0 ≤ x ≤ 3, find the cumulative distribution function F(x) for 0 ≤ x ≤ 3, and evaluate F(1.5).
11.A random variable has cumulative distribution function F(x) = for 0 ≤ x ≤ 4. Find its density function and P(1 < X < 3).
12.For the distribution with F(x) = on 0 ≤ x ≤ 4, find the median, correct to 2 decimal places. (answer space)
Preview — the PDF contains the full resource and answer sheet.
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About this resource
- Created by
- Success Tutoring
- Last reviewed
- 1 October 2026
- How it was made
- Written by our team
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