Question

2.1.5 Let A and B be events, and let X = 1A" 1B. Is X an indicator function? If yes, then of what event?

Answer 1

A, B events X = I_A middot I_B PART a) from RHS = I_A middot I_B 4 cases x elementof A, x notelementof B I_A middot I_B = 1.0 = 0 x notelementof A, x elementof B I_A middot I_B = 0.1 = 0 x notelementof A, x notelementof B I_A middot I_B = 0 middot 0 = 0 x elementof A, x elememtof B I_A middot I_B = 1 middot 1 = 1 Therefore takes value 1,0 X = 1 when x elementof A and x elementof B X = 0 else Therefore X is a characteristic function which is X = {1 x elementof A intersection B 0 x notelementof A intersection B X = I_A intersection B \r\n PART - b To prove: I_A intersection B = min {I_A, I_B} if x elementof A intersection B, then I_A intersection B = 1 and x elementof A and x elementof B