18.05 Lecture Notes - Lecture 4: Mit Opencourseware
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P (x x) = 1 p (x < x) = 1 x. E xdx = 1 (1 e x) = e x . (b) for t 0, we know that t t if and only if both x1 t and x2 t. so. P (t t) = p (x1 t, x2 t). P (x1 t, x2 t) = p (x1 t)p (x2 t) = e 2 t . Thus, ft (t) = p (t t) = 1 e 2 t . Di erentiating with respect to t to get the pdf, we nd ft (t) = ft. T is an exponential random variable with mean. 2 (c) let x1, x2, and x3 be the lifetimes of bulbs b1, b2 and b3, respectively. Then we know x1 exp(2), x2 exp(3), x3 exp(5). Then t is the time to the rst failure of a bulb. Following the same argument as in (b), we have.
