2. Let X and Y be continuous random variables having the joint density function: [c(x² + y²) 0 f(x, y) = 0≤x≤1, 0≤y≤1 otherwise Determine a) The constant c b) P(X< 1/2, Y> 1/2) c) P(1/4 < X < 3/4) d) P(Y<1/2) e) The conditional distribution of X given Y=y f) Whether or not X and Y are independent.
2. Let X and Y be continuous random variables having the joint density function: [c(x² + y²) 0 f(x, y) = 0≤x≤1, 0≤y≤1 otherwise Determine a) The constant c b) P(X< 1/2, Y> 1/2) c) P(1/4 < X < 3/4) d) P(Y<1/2) e) The conditional distribution of X given Y=y f) Whether or not X and Y are independent.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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a. The constant c is 3/2
Only need solutions for e,f
![**Problem Statement:**
2. Let \( X \) and \( Y \) be continuous random variables having the joint density function:
\[
f(x, y) =
\begin{cases}
c(x^2 + y^2) & \text{if } 0 \leq x \leq 1, \ 0 \leq y \leq 1 \\
0 & \text{otherwise}
\end{cases}
\]
Determine:
a) The constant \( c \)
b) \( P(X < 1/2, Y > 1/2) \)
c) \( P(1/4 < X < 3/4) \)
d) \( P(Y < 1/2) \)
e) The conditional distribution of \( X \) given \( Y = y \)
f) Whether or not \( X \) and \( Y \) are independent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F832196dd-2d45-47e5-91ed-72a7273417b1%2Fd1b4ee89-309f-4243-b38f-2d11d0536304%2F3ekieyt_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
2. Let \( X \) and \( Y \) be continuous random variables having the joint density function:
\[
f(x, y) =
\begin{cases}
c(x^2 + y^2) & \text{if } 0 \leq x \leq 1, \ 0 \leq y \leq 1 \\
0 & \text{otherwise}
\end{cases}
\]
Determine:
a) The constant \( c \)
b) \( P(X < 1/2, Y > 1/2) \)
c) \( P(1/4 < X < 3/4) \)
d) \( P(Y < 1/2) \)
e) The conditional distribution of \( X \) given \( Y = y \)
f) Whether or not \( X \) and \( Y \) are independent.
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