A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 6: Q 6.22 (page 272)

The joint density function of $X$ and $Y$ is
$f (x, y) = \{\begin{cases} x + y 鈥呪赌呪赌呪赌� 0 < x < 1, 0 < y < 1 \\ 0 鈥呪赌呪赌呪赌� otherwise \end{cases}$
(a) Are $X$ and $Y$ independent?
(b) Find the density function of $X$ .
(c) Find $P {X + Y < 1}$ .

Short Answer

Expert verified

The variables X and Y are dependent.
The density function of random variable X is:

$f_{X} (x) = x + \frac{1}{2}; 0 < x < 1$

$c . P (X + Y < 1) = \frac{1}{3}$

Step by step solution

Step 1. Given information:

$f (x, y) = \{\begin{matrix} x + y; 0 < x < 1, 0 < y < 1 \\ 0;otherwise \end{matrix}\}$

Step 2. Prove the independence of X and Y.

In order to test the independence of X and Y, we need to find the marginal distribution function of X and Y. Therefore,

$\begin{matrix} f_{X} (x) = \underset{y}{鈭�} f (x, y) d y \\ = {鈭�}_{0}^{1} (x + y) d y \\ = {(x y + \frac{y^{2}}{2})}_{0}^{1} \\ = x + \frac{1}{2} \\ f_{X} (x) = x + \frac{1}{2}; 0 < x < 1 \end{matrix}$

Since the variables $X$ and $Y$ are interchangeable/symmetric, therefore,

$f_{Y} (y) = y + \frac{1}{2}; 0 < y < 1$

Since $f (x, y) 鈮� f_{X} (x) f_{Y} (y)$

Therefore, $X$ and $Y$ are dependent random variables.

Density function of X

From the above step, we get-

$f_{X} (x) = x + \frac{1}{2}; 0 < x < 1$

Calculation of P(X+Y<1)

$\begin{array}{l} P (X + Y < 1) = \underset{x}{鈭�} \underset{y}{鈭�} f (x, y) d x d y \\ = {鈭�}_{x = 0}^{1} {鈭�}_{y = 0}^{1 鈭� x} (x + y) d x d y \\ = {鈭�}_{0}^{1} {|x y + \frac{y^{2}}{2}|}_{0}^{1 鈭� x} d x \\ = {鈭�}_{0}^{1} [x (1 鈭� x) + \frac{{(1 鈭� x)}^{2}}{2}] d x \\ = {鈭�}_{0}^{1} (\frac{1 鈭� x^{2}}{2}) d x \\ = {|\frac{3 x 鈭� x^{3}}{6}|}_{0}^{1} \\ 鈬� P (X + Y < 1) = \frac{1}{3} \end{array}$

which is the required solution.

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91影视

The joint density function of $X$ and $Y$ is
$f (x, y) = \{\begin{cases} x + y 鈥呪赌呪赌呪赌� 0 < x < 1, 0 < y < 1 \\ 0 鈥呪赌呪赌呪赌� otherwise \end{cases}$
(a) Are $X$ and $Y$ independent?
(b) Find the density function of $X$ .
(c) Find $P {X + Y < 1}$ .

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Prove the independence of X and Y.

Density function of X

Calculation of P(X+Y<1)

One App. One Place for Learning.

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91影视

The joint density function of Xand Yisf(x,y)=x+y鈥呪赌呪赌呪赌�0<x<1,0<y<10鈥呪赌呪赌呪赌�otherwise(a) Are Xand Yindependent?(b) Find the density function of X.(c) FindP{X+Y<1}.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Prove the independence of X and Y.

Density function of X

Calculation of P(X+Y<1)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Statistics

Pure Maths

Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

The joint density function of $X$ and $Y$ is
$f (x, y) = \{\begin{cases} x + y 鈥呪赌呪赌呪赌� 0 < x < 1, 0 < y < 1 \\ 0 鈥呪赌呪赌呪赌� otherwise \end{cases}$
(a) Are $X$ and $Y$ independent?
(b) Find the density function of $X$ .
(c) Find $P {X + Y < 1}$ .