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Chapter 8: Q 8.4 (page 390)
Let X1, ... , X20 be independent Poisson random variables with mean 1.
(a) Use the Markov inequality to obtain a bound on
(b) Use the central limit theorem to approximate
a)
b)
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Compute the measurement signal-to-noise ratio that is, |渭|/蟽, where 渭 = E[X] and 蟽2 = Var(X) of the
following random variables:
(a) Poisson with mean 位;
(b) binomial with parameters n and p;
(c) geometric with mean 1/p;
(d) uniform over (a, b);
(e) exponential with mean 1/位;
(f) normal with parameters 渭, 蟽2.
Let be a Poisson random variable with a mean of.
Use the Markov inequality to obtain an upper bound.
Use the one-sided Chebyshev inequality to obtain an upper bound.
Use the Chernoff bound to obtain an upper bound.
Approximate by making use of the central limit theorem.
Determine by running an appropriate program.
A clinic is equally likely to have 2, 3, or 4 doctors volunteer for service on a given day. No matter how many volunteer doctors there are on a given day, the numbers of patients seen by these doctors are independent Poisson random variables with a mean of . Let X denote the number of patients seen in the clinic on a given day.
(a) Find
(b) Find Var
(c) Use a table of the standard normal probability distribution to approximate P.
Use the central limit theorem to solve part of the problemlocalid="1649757874152" .
8.5 The amount of time that a certain type of component functions before failing is a random variable with probability density function
Once the component fails, it is immediately replaced by
another one of the same type. If we let denote the life-time of the th component to be put in use, then represents the time of the th failure. The long-term rate at which failures occur, call it, is defined by
Assuming that the random variables are independent, determine .
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