Solution: Probability A joint density function of the continuous random variables and is

Chapter 14, Problem 65

(choose chapter or problem)

Probability A joint density function of the continuous random variables x and y is a function f(x, y) satisfying the following properties.

(a) \(f(x, y) \geq 0\) for all (x, y)

(b) \(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) d A=1\)

(c) \(P[(x, y) \in R]=\int_{R} \int f(x, y) d A\)

In Exercises 63 - 66, show that the function is a joint density function and find the required probability.

\(f(x, y)= \begin{cases}\frac{1}{27}(9-x-y), & 0 \leq x \leq 3,3 \leq y \leq 6 \\ 0, & \text { elsewhere }\end{cases}\)

\(P(0 \leq x \leq 1,4 \leq y \leq 6)\)

Text Transcription:

f(x, y) geq 0

int_{-infty}^{infty} int_{-infty}^{infty} f(x, y) d A = 1

P[(x, y) in R] = int_{R} int f(x, y) d A

f(x, y) = {1 / 27 (9 - x - y),  0 leq x leq 3, 3 leq y leq 6 \\ 0,  elsewhere

P(0 leq x leq 1, 4 leq y leq 6)