# 2 Discrete Random Variable. Independence (module 1) A and B are independent iff the knowledge that B has occurred does not change the probability that.

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2 Discrete Random Variable

Independence (module 1) A and B are independent iff the 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 S X = {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[M  k+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 6 Megabit per second or higher?

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