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Question: 5 45 points exponential random variable let x be an...

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5. (45 points] Exponential Random Variable. Let X be an exponentially distributed random variable with rate A. (a) [5 point] Derive the density: (x) (b) (5 point] Show that P (X> ) eAr (Hint: no need to do any integrals.) e) [10 pointsj Show that E X] 1/. (Hint: use integration by parts.) (d) [10 points) Show that X satisfies the inemoryless property (e) [15 points) Suppose X represents the lifetime of a lightbulb and E X] 10 hours (so 0.1 per hour). That is, the lightbulb will burn out after being used for X hours. At 10 hours, the lightbulb is still burning, what is the probability that it will last for the next 10 hours? Hint: Use the memoryless property.
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