Since the Q- function represents the tail probability of a Gaussian random variable, we can use the various bounds on tail probabilities to produce bounds on the Q- function.
(a) Use Markov’s inequality to produce an upper bound on the Q- function. Hint: a Gaussian random variable has a two- sided PDF, and Markov’s inequality requires the random variable to be one- sided. You will need to work with absolute values to resolve this issue.
(b) Use Chebyshev’s inequality to produce an upper bound on the Q- function.
(c) Plot your results from parts (a) and (b) along with the bound obtained from the Chernoff bound from Example 4.28. In order to determine how tight (or loose) these various bounds are, also include on your plot the exact value of the Q- function.