---
title: "Joint pdf a(x+y): marginals, probability"
subject: "Probability and Statistics"
chapter: 3
key: "semester_5/probability_and_statistics:joint-pdf-a-x-plus-y"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/joint-pdf-a-x-plus-y
site: BCT Engineering
---

# Joint pdf a(x+y): marginals, probability

Probability and Statistics · Chapter 3: Continuous Probability Distributions

- Asked in: 2068 Jestha (old course), 6 marks
- Video: [Marginal PDF from Joint PDF](https://www.youtube.com/watch?v=tcEjpVLYEjc) — math et al
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/joint-pdf-a-x-plus-y
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3

## Question

The given joint probability density function is f(x, y) = a(x + y), for 0 < x < 1 and 1 < y < 2; = 0 elsewhere.
i) Find the marginal function of x and y.
ii) Find the probability for (0.5 < x < 0.8 and y > 1.5).

## Short answer

a=1/2. Marginal of X: g(x)=(2x+3)/4 for 0<x<1. Marginal of Y: h(y)=(2y+1)/4 for 1<y<2. P(0.5<X<0.8, Y>1.5)=0.18.

## Answer

#### Find the constant a
For a joint pdf, ∫∫ f(x, y) dy dx = 1.

∫₀¹ ∫₁² a(x + y) dy dx = a ∫₀¹ [xy + y²/2]₁² dx = a ∫₀¹ (x + 3/2) dx = a [x²/2 + 3x/2]₀¹ = a(1/2 + 3/2) = 2a

2a = 1, so **a = 1/2**. Thus f(x, y) = (x + y)/2, 0 < x < 1, 1 < y < 2.

#### i) Marginal functions
Marginal of X:

g(x) = ∫₁² (x + y)/2 dy = (1/2) [xy + y²/2]₁² = (1/2)(x + 3/2) = **(2x + 3)/4**, 0 < x < 1

Marginal of Y:

h(y) = ∫₀¹ (x + y)/2 dx = (1/2) [x²/2 + xy]₀¹ = (1/2)(1/2 + y) = **(2y + 1)/4**, 1 < y < 2

Check: ∫₀¹ (2x + 3)/4 dx = (1 + 3)/4 = 1 and ∫₁² (2y + 1)/4 dy = (3 + 1)/4 = 1.

#### ii) P(0.5 < X < 0.8 and Y > 1.5)
P = ∫₀.₅^₀.₈ ∫₁.₅² (x + y)/2 dy dx

Inner integral: ∫₁.₅² (x + y) dy = [xy + y²/2]₁.₅² = 0.5x + (2 − 1.125) = 0.5x + 0.875

P = (1/2) ∫₀.₅^₀.₈ (0.5x + 0.875) dx = (1/2) [0.25x² + 0.875x]₀.₅^₀.₈

= (1/2) [(0.16 + 0.7) − (0.0625 + 0.4375)] = (1/2)(0.86 − 0.50) = 0.18

**Answer: a = 1/2; g(x) = (2x + 3)/4; h(y) = (2y + 1)/4; P = 0.18.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The double integral of a(x + y) over 0<x<1, 1<y<2 is 2a, so a is:
   A. 2
   B. 1/2
   C. 1
   D. 1/4
   Answer: B. A joint pdf integrates to 1, so 2a = 1 and a = 1/2. (Key point: Finding a)

2. The marginal function of X is obtained by:
   A. integrating f(x, y) over y
   B. integrating f(x, y) over x
   C. differentiating f
   D. multiplying by y
   Answer: A. g(x) = ∫ f(x, y) dy over the range of y, giving (2x + 3)/4. (Key point: Marginal of X)

3. The marginal function of Y is:
   A. (2x + 3)/4
   B. (2y + 1)/4
   C. (y + 1)/2
   D. (x + y)/2
   Answer: B. h(y) = ∫0^1 (x + y)/2 dx = (1/2)(1/2 + y) = (2y + 1)/4. (Key point: Marginal of Y)

4. P(0.5 < X < 0.8 and Y > 1.5) is a:
   A. double integral over the region
   B. product of the marginal means
   C. single integral of g(x) only
   D. sum of the marginals
   Answer: A. Integrate the joint pdf over x from 0.5 to 0.8 and y from 1.5 to 2, giving 0.18. (Key point: Joint probability)

## Related in Probability and Statistics

- [Joint pmf: marginals and independence](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Ajoint-pmf-marginal-independence) — Probability and Statistics, chapter 2
- [Continuous variable with density 6x(1−x)](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apdf-6x-1-minus-x) — Probability and Statistics, chapter 3
- [pdf Ax(6−x)²: find A, mean, variance](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apdf-ax-6-minus-x-squared) — Probability and Statistics, chapter 3, asked 2×
- [pdf definition; f(x)=Kx(1−x) probabilities](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apdf-kx-1-minus-x-probabilities) — Probability and Statistics, chapter 3
- [pdf cx²: constant, P(1<x<2), cdf](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apdf-cx2-0-3) — Probability and Statistics, chapter 3
- [Verify pdf 2e^(−2x); P(1<x<3); mean](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aexponential-pdf-2e-2x-verify) — Probability and Statistics, chapter 3

## Related in other subjects

- [Fermi-Dirac function and 1.5kT probability](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Afermi-dirac-1-5kt) — Electrical Engineering Material, chapter 1
- [Valid lossless function; Foster II and Cauer II](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Alossless-three-functions-foster-cauer-ii) — Filter Design, chapter 4, asked 2×
- [Truss joint D: loading, lack of fit, temperature](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-defl-d-2074) — Theory of Structures I, chapter 3, asked 3×
- [Truss joint E: loading, lack of fit, temperature](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-defl-e-2068) — Theory of Structures I, chapter 3, asked 2×