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![1. Use a software to simulate 1,000 random numbers from an Exponential distribution whose mean is 50,<br>i.e., X - Exp(50). Verify the memory-less property of the Exponential distribution using your simulated<br>data through the following steps:<br>(a) Estimate P(X > t) for t = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50.<br>(b) Estimate P(X > t + 10 | X > 10) for t = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. Plot the estimated<br>probabilities in (b) against those in (a) in one graph.<br>(c) Estimate P(X > t + 20 | X > 20) for t = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. Plot the estimated<br>probabilities in (c) against those in (b) in one graph.<br>(d) Summarize your findings.<br>2. Round up the above simulated random number to the smallest integer that is no less than the random<br>number, i.e., let Y = [X]. For example, [0.01] = 1, [1.0] = 1, and [5.62] = 6. Repeat the four parts in<br>Problem 1 using the transformed data set. What is this distribution?<br>](https://s3.us-east-1.amazonaws.com/storage.unifolks.com/qimg-008/008_jsb7yj0-mnhuvc0m.jpeg)
Extracted text: 1. Use a software to simulate 1,000 random numbers from an Exponential distribution whose mean is 50, i.e., X - Exp(50). Verify the memory-less property of the Exponential distribution using your simulated data through the following steps: (a) Estimate P(X > t) for t = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. (b) Estimate P(X > t + 10 | X > 10) for t = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. Plot the estimated probabilities in (b) against those in (a) in one graph. (c) Estimate P(X > t + 20 | X > 20) for t = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. Plot the estimated probabilities in (c) against those in (b) in one graph. (d) Summarize your findings. 2. Round up the above simulated random number to the smallest integer that is no less than the random number, i.e., let Y = [X]. For example, [0.01] = 1, [1.0] = 1, and [5.62] = 6. Repeat the four parts in Problem 1 using the transformed data set. What is this distribution?