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5 pair of RVs

5 pair of RVs. 5-1: joint pmf. In a box are three dice. Die 1 is normal; die 2 has no 6 face, but instead two 5 faces; die 3 has no 5 face, but instead two 6 faces. The experiment consists of selecting a die at random, followed by a toss with that die.

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5 pair of RVs

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  1. 5 pair of RVs

  2. 5-1: joint pmf • In a box are three dice. Die 1 is normal; die 2 has no 6 face, but instead two 5 faces; die 3 has no 5 face, but instead two 6 faces. • The experiment consists of selecting a die at random, followed by a toss with that die. • Let X be the die number that is selected, and let Y be the face value of that die. • Find P(X = x, Y = y) for all possible x and y.

  3. 5-2: Joint pmf • A packet switch has two input ports and two output ports. At a given time slot, a packet arrives at each input port with probability and is equally likely to be destined to output port 1 or 2. Let X and Y be the number of packets destined for output ports 1 and 2, respectively. Find the pmf of X and Y. • Three outcomes for an input port can take the following values: (i) “n”, no packet arrival; (ii) “a1”, packet arrival destined for output port 1; (iii) “a2”, packet arrival destined for output port 2.

  4. 5-3: marginal pmf • We pick a message, whose length N follows a geometric distribution with parameter 1-p and SN={0,1,2,…}. • Find the joint pmf and the marginal pmf’s of Q and R, where Q is the quotient in the division of N by constant M, and R is the number of remaining bytes.

  5. 5-4: joint cdf

  6. 5-5: joint pdf • Joint pdf is given by: • Find c. • Find marginal pdf’s • Find P[X+Y1]

  7. 5-6: independence

  8. 5-7: condi. prob. (discrete) • The total number of defects X on a chip is a Poisson random variable with mean . • Each defect has a probability p of falling in a specific region R and the location of each defect is independent of the locations of other defects. • Find the pmf of the number of defects Y that fall in the region R.

  9. 5-8: condi. Prob. (conti.) • X is selected at random from the unit interval; Y is then selected at random from the interval (0, X). • Find the cdf of Y.

  10. 5-9: function of RVs • A system with standby redundancy has a single key component in operation and a duplicate of that component in standby mode. • When the first component fails, the second component is put into operation. • Find the pdf of the lifetime of the standby system if the components have independent exponentially distributed lifetimes with the same mean .

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