Question

Suppose X and Y are independent and identically distributed random variables that are distributed uniformly in the interval [0,1]. The probability that X≥Y is_____.

Hint:

For uniform distributions over a square, probabilities involving inequalities correspond to geometric areas. Use symmetry or simple integration for efficient calculation.

Answer 1

Correct Answer: 0.5

To find the probability that \( X \geq Y \) for two independent and identically distributed uniform random variables \( X \) and \( Y \) over the interval \([0,1]\), we approach this problem geometrically. The joint distribution of \( X \) and \( Y \) is uniform over the unit square defined by \([0,1] \times [0,1]\). The event \( X \geq Y \) corresponds to the region in this square where the \( x \)-coordinate is greater than or equal to the \( y \)-coordinate. Graphically, this region is a triangle bounded by the line \( x = y \) from \((0,0)\) to \((1,1)\) and the x-axis from \((0,0)\) to \((1,0)\), and the line \( x = 1 \) from \((1,0)\) to \((1,1)\). The area of the triangle where \( X \geq Y \) can be calculated. The triangle has a base and height of length 1 (the sides of the unit square), thus: \[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}.\] This area represents the probability that \( X \geq Y \), as the joint distribution is uniform. Therefore, the probability that \( X \geq Y \) is \( 0.5 \). Since the problem provides a range as \( 0.5,0.5 \), the computed probability of \( 0.5 \) indeed fits within the specified range.