propability theory
Appendix D
Additional problems
D.1 Probability theory (Chapter 2-3)
(i) If X is a uniform, continuous random variable on the interval [a, b] and Y is a uniform, discrete random variable on the interval [k, l] where k and l are integers and k < l, then compute Pr [Z ≤ x] where
Z = X + Y given that X and Y are independent.
(ii) Suppose that the number N of pages in a fax transmission has a geometric probability distribution with mean 1/q = 4. The number
K of bits per page also has a geometric distribution with mean 1/p =
105 bits, independent of any other page and of the number of pages.
Show that the total number B of bits in a fax transmission is a geometric random variable with parameter pq.
(iii) Random variables whose moments are also random variables. Compute the probability density function of the random variable X that has (a) a lognormal distribution with zero mean where the variance is exponentially distributed with mean λ−1 .
(b) a Poisson distribution whose mean λ is exponentially distributed with mean μ.
(iv) Let X denote the number of packets in a flow F and let Y denote the number of packets that are sampled from that original flow F . The conditional distribution of Y , given that X = l, follows a binomial distribution. What is the probability that a sampled flow of length k is sampled from an original flow of length l? Original flow lengths are geometrically distributed.
(v) Uniform sampling. Consider a set N of size N that contains items of type A and type B. The subset with m items of type A is denoted by
M. A uniform sampling strategy consists of testing the type of an
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546
Additional problems
arbitrary item of N . Once tested, the item is not replaced in the set
N . Show that each drawing/test in the uniform sampling strategy has precisely probability p = m to find an item of type A. An example
N
is the test of infected hosts in the Internet by randomly chosing an
IP address and testing whether the host
Additional problems
D.1 Probability theory (Chapter 2-3)
(i) If X is a uniform, continuous random variable on the interval [a, b] and Y is a uniform, discrete random variable on the interval [k, l] where k and l are integers and k < l, then compute Pr [Z ≤ x] where
Z = X + Y given that X and Y are independent.
(ii) Suppose that the number N of pages in a fax transmission has a geometric probability distribution with mean 1/q = 4. The number
K of bits per page also has a geometric distribution with mean 1/p =
105 bits, independent of any other page and of the number of pages.
Show that the total number B of bits in a fax transmission is a geometric random variable with parameter pq.
(iii) Random variables whose moments are also random variables. Compute the probability density function of the random variable X that has (a) a lognormal distribution with zero mean where the variance is exponentially distributed with mean λ−1 .
(b) a Poisson distribution whose mean λ is exponentially distributed with mean μ.
(iv) Let X denote the number of packets in a flow F and let Y denote the number of packets that are sampled from that original flow F . The conditional distribution of Y , given that X = l, follows a binomial distribution. What is the probability that a sampled flow of length k is sampled from an original flow of length l? Original flow lengths are geometrically distributed.
(v) Uniform sampling. Consider a set N of size N that contains items of type A and type B. The subset with m items of type A is denoted by
M. A uniform sampling strategy consists of testing the type of an
545
546
Additional problems
arbitrary item of N . Once tested, the item is not replaced in the set
N . Show that each drawing/test in the uniform sampling strategy has precisely probability p = m to find an item of type A. An example
N
is the test of infected hosts in the Internet by randomly chosing an
IP address and testing whether the host