Solution Found!
For each of the following functions, determine the constant c so that f(x, y) satisfies
Chapter 4, Problem 4.1-1(choose chapter or problem)
For each of the following functions, determine the constant \(c\) so that \(f(x, y)\) satisfies the conditions of being a joint pmf for two discrete random variables \(X\) and \(Y\):
(a) \(f(x, y)=c(x+2 y), \quad x=1,2, \quad y=1,2,3\).
(b) \(f(x, y)=c(x+y), \quad x=1,2,3, \quad y=1, \ldots, x\)
(c) \(f(x, y)=c, \quad x\) and \(y\) are integers such that \(6 \leq x+y\) \(\leq 8,0 \leq y \leq 5\) .
(d) \(f(x, y)=c\left(\frac{1}{4}\right)^{x}\left(\frac{1}{3}\right)^{y}, \quad x=1,2, \ldots, \quad y=1,2, \ldots\)
Questions & Answers
QUESTION:
For each of the following functions, determine the constant \(c\) so that \(f(x, y)\) satisfies the conditions of being a joint pmf for two discrete random variables \(X\) and \(Y\):
(a) \(f(x, y)=c(x+2 y), \quad x=1,2, \quad y=1,2,3\).
(b) \(f(x, y)=c(x+y), \quad x=1,2,3, \quad y=1, \ldots, x\)
(c) \(f(x, y)=c, \quad x\) and \(y\) are integers such that \(6 \leq x+y\) \(\leq 8,0 \leq y \leq 5\) .
(d) \(f(x, y)=c\left(\frac{1}{4}\right)^{x}\left(\frac{1}{3}\right)^{y}, \quad x=1,2, \ldots, \quad y=1,2, \ldots\)
ANSWER:
Step 1 of 5
Given function is
\(f(x, y)=c(x+2 y), \quad x=1,2, \quad y=1,2,3\)
To find the constant \(c\) such that \(f(x, y)\) satisfies the condition of being a joint pmf for two discrete random variables \(X\) and \(Y\)
For the function \(f(x, y)\) to satisfy the condition of being a joint pmf for two discrete random variables \(X\) and \(Y, f(x, y)\) must satisfy the following properties:-
(1) \(0 \leq f\left(x_{i}, y_{i}\right) \leq 1\)
(2) \(\sum_{i=1}^{n} \sum_{j=1}^{m} f\left(x_{i}, y_{j}\right)=1\)