两独立 U(0,1)： $XY>1/2$ 概率

乘积大于 1/2

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

Suppose we draw two random numbers $X$ and $Y$ each distributed uniform on the interval $[0, 1]$ . If $X$ and $Y$ are independent, what is the probability that their product is greater than $1 / 2$ ?

解析

在单位正方形上积分。条件 $XY>1/2$ 等价于 $y>\\frac{1}{2x}$ 且 $x\\ge 1/2$ 。\n\n面积为\n\n $\n\\int_{1/2}^{1}\\left(1-\\frac{1}{2x}\\right)dx=\\left[x-\\frac{1}{2}\\ln x\\right]_{1/2}^{1}=\\frac{1}{2}(1-\\ln 2).\n$ \n\n因此\n\n $\n\\boxed{\\mathbb{P}(XY>1/2)=\\frac{1}{2}(1-\\ln 2)\\approx 0.1534}.\n$