1. consider two jointly discrete random variables with joint probability mass function given…

1. consider two jointly discrete random variables with joint probability mass function given below\nfind:\n(a) f(2,3)\n(b) f(-1,2)\n(c) p(x + y <= 2)

1. consider two jointly discrete random variables with joint probability mass function given below\nfind:\n(a) f(2,3)\n(b) f(-1,2)\n(c) p(x + y <= 2)

Answer

Explanation:

Step1: 计算 $F(2,3)$

联合累积分布函数 $F(x,y) = P(X \le x, Y \le y)$。由于所有概率之和为 1，且 $x=2, y=3$ 涵盖了所有可能的取值，故 $F(2,3) = 1$。

Step2: 计算 $F(-1,2)$

由于 $X$ 的最小取值为 0，当 $x < 0$ 时，$P(X \le -1, Y \le 2) = 0$。

Step3: 计算 $P(X+Y \le 2)$

求满足 $x+y \le 2$ 的所有点 $(x,y)$ 的概率之和： $P(0,0) + P(0,1) + P(0,2) + P(1,0) + P(1,1) + P(2,0)$ $= \frac{1}{9} + \frac{1}{9} + \frac{1}{9} + \frac{1}{9} + \frac{1}{9} + \frac{1}{9} = \frac{6}{9} = \frac{2}{3}$

Answer:

(a) $F(2,3) = 1$ (b) $F(-1,2) = 0$ (c) $P(X+Y \le 2) = \frac{2}{3}$