Probability A joint density function of the continuous randomvariables and is a function

Chapter 14, Problem 74

(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 73-76, show that the function is a joint density function and find the required probability.

\(f(x, y)=\left\{\begin{array}{ll} \frac{1}{4} x y, & 0 \leq x \leq 2,0 \leq y \leq 2 \\ 0, & \text { elsewhere } \end{array}\right.\)

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

Text Transcription:

f(x, y) geq 0

int_{-infty}^{infty} int_{-infty}^{infty} f(x, y)

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

f(x, y) = {_0, elsewhere ^1/4 xy, 0 leq x leq 2, 0 leq y leq 2

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