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## 2 Discrete Random Variable

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**Independence (module 1)**• A and B are independent iffthe knowledge that B has occurred does not change the probability that A occurs. • A, B independent ⇔ P[A|B] = P[A] • A, B independent ⇔ P(A B) = P(A) P(B)**Random Variable (RV)**• 2-1: Throw two dice. An RV X maps the sum of the numbers of two dice from the experiment. Describe the function X. • 2-2: Toss a coin three times. If the number of heads, X, is 3, the reward is 8; if X = 2, the reward is 2; otherwise, 0. Let Y be the RV for the reward. Describe Y.**Prob. mass function (pmf)**• 2-3: Find the pmf of X, which is the label of a ball picked at random from an urn that contains balls labeled from 1 to 5. • 2-4: Find the pmf of X, which is the number of largest face when two dice are rolled.**expectation**• 2-5: expectation of X, which is number of heads in three tosses of a coin • 2-6: the average value when we pick an integer between 1 and M at random (uniformly).**variance**• 2-7: variance of X, which is the number of heads in two tosses of a coin • 2-8: variance of a Bernoulli RV.**Conditional Prob.**• 2-9: Let X be the time required to transmit a message, where X is a (discrete) uniform RV with SX = {1, 2, …, L}. Suppose that a message has already been transmitted for m time units, find the probability that the remaining transmission time is j time units.**2-10: Binomial RV**• A system uses triple redundancy for reliability. Three microprocessors are installed and the system is designed so that it operates as long as one microprocessor is still functional. • Suppose that the probability that a microprocessor is still active after t seconds is p = e-t. • Find the probability that the system is still operating after t seconds.**HW 2-1 Geometric RV**• RV M is the number of tosses of a coin until the head first appears • Compare two probabilities: P[Mk+j| M>j] and P[M k] • Think about the comparison result**HW 2-2 Poisson RV**• At a given time, the number of households connected to the Internet is a Poisson RV with mean 3. Suppose that the transmission bit rate available for all the households is 20 Megabits per second. • Find the probability of the distribution of the transmission bit rate per user. • If no household is connected, the rate is • Find the transmission bit rate that is available to a user with probability 90% or higher. • What is the probability that a user has a share of 6Megabit per second or higher?