Probability Theory and Statistics

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Probability Theory and Statistics

  1. X is the Uniform(−5, 5) random variable. (a) Whatis fX(x)?
    (b) WhatisFX(x)?
    (c) What is E[X]?
    (d) WhatisE[eX]?
    (e) WhatisP(|X|<σX)?
  2. The amount of time required for your computer to connect to a router is an exponential random variable, with mean connection time of 3.0 seconds.
    1. (a) What is the probability that a connection will be made in 5.0 seconds?
    2. (b) If, after 5.0 seconds there is still no connection, what is the probability the connection will be made in the following 5.0 seconds (so, within 10.0 seconds total)?
    3. (c) Compare your answers to (a) and (b) and explain the observed behavior of this random variable.
  3. The lifetimes of radioactive nuclei are independent and exponentially distributed. The half-life is given by t1/2 = ln 2/λ, where t1/2 is the time such that the expected fraction of atoms remaining is 1/2.
    1. (a) Iodine-131 is radioactive, with a half-life of 8.02 days. What is the probability that a single atom of Iodine-131 is still un-decayed after 30 days?
    2. (b) How long do you have to wait before 99.99% of a sample of Iodine-131 has decayed?
    3. (c) Uranium-238 is radioactive, with a half-life of 4.468 × 109 years. Suppose you have an 0.1 mg sample of Uranium-238. Let X be the number of nuclei that decay in one second in your sample; calculate E[X]. What is P(X ≥ 2)?
  4. Continuous random variable X has E[X] = 3 and Var(X) = 9. Find the parameters of X if (a) X is an exponential random variable
    (b) X is an Erlang random variable
    (c) X is a continuous uniform random variable
  5. If X is Exponential(λ) and W = X2, what is the PDF of W?