# Question

Let Xk be a sequence of IID exponential random variables with mean of 1. We wish to compute

For some constant y (such that y > 25).

(a) Find a bound to the probability using Markov’s inequality.

(b) Find a bound to the probability using Chebyshev’s inequality.

(c) Find a bound to the probability using the Chernoff bound.

(d) Find an approximation to the probability using the central limit theorem.

(e) Find the exact probability.

(f) Plot all five results from (a) through ( e) for y = 25 and determine for what range of y the central limit theorem gives the most accurate approximation compared with the 3 bounds.

For some constant y (such that y > 25).

(a) Find a bound to the probability using Markov’s inequality.

(b) Find a bound to the probability using Chebyshev’s inequality.

(c) Find a bound to the probability using the Chernoff bound.

(d) Find an approximation to the probability using the central limit theorem.

(e) Find the exact probability.

(f) Plot all five results from (a) through ( e) for y = 25 and determine for what range of y the central limit theorem gives the most accurate approximation compared with the 3 bounds.

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