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. 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.

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 ½ 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 joint pmf of (X,Y). Three outcomes for each 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.

5-3: marginal pmf We pick a message, whose length N follows a geometric distribution with parameter 1-p and SN={0,1,2,…}. 𝑃 𝑁=𝑘 = 𝑝 𝑘 (1−𝑝) 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.

5-4: joint cdf Joint cdf of (X,Y) is given by 𝐹 𝑋,𝑌 𝑥,𝑦 = 1− 𝑒 −𝑥 1− 𝑒 −𝛽𝑦 𝑖𝑓 𝑥, 𝑦≥0 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Find marginal cdf of X and Y Find P[X<1, Y<1] Find P[X>x, Y>y] Find P[1<X<2, 2<Y<5}

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

5-6: independence Check whether the following joint pdf is independent or not 𝑓 𝑋,𝑌 (𝑥,𝑦)= 9𝑥 2 𝑦 2 if 0x,y 1

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.

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.

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 .