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

Chapter 14, Problem 63

(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.

\(f(x, y)=\left\{\begin{aligned} \frac{1}{10}, & 0 \leq x \leq 5,0 \leq y \leq 2 \\ 0, & \text { elsewhere } \end{aligned}\right\).

\(P(0 \leq x \leq 2,1 \leq y \leq 2)\)

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 / 10,  0 leq x leq 5, 0 leq y leq 2 \\ 0,  elsewhere

P(0 leq x leq 2,1 leq y leq 2