4. A black and white (monochrome) CRT monitor uses a single electron gun to create images. A color monitor uses three electron guns (red, green, blue) to create the same quality images, but in color. Dale-Marie buys a monochrome CRT monitor, and Dan buys a color monitor both manufactured by the same company with the same lifetime distribution for the electron guns. Suppose that each electron gun lasts, independently of other electron guns, for an exponential period with parameter λ . Naturally, Dale-Marie will discard her monitor when it does not produce images; that is, when the electron gun of the monitor dies. Dan discards his monitor when it malfunctions; that is, when for the first time one of the electron guns of his monitor dies. Suppose that Dale-Marie’s monitor works for X units of time, and Dan’s monitor will be discarded after working for Y units of time. Find P(X < Y ), E



max(X, Y )



, and E



min(X < Y )



.

Hint: Let Y1, Y2, and Y3 be the lifetimes of the electron guns of Dan’s monitor. Note that Y = min(Y1, Y2, Y3).