ECE270: Handout 6Discrete RVs (Part I)Outline:1. Bernoulli RV, its pmf and CDF,2. Binomial RV, its pmf and CDF,3. Geometric RV, its pmf and CDF,4. Poisson RV, its pmf and CDF,5. Discrete uniform RV, its pmf and CDF.F Important Discrete RVs• Certain discrete RVs arise in many applications (engineering, Physics, biology, economics,...), due to the fact that they model fundamental mechanism that underlie random behav-ior. Important discrete RVs are Bernoulli RV,Binomial RV, Geometric RV, Poisson RV, anddiscrete uniform RV.F Bernoulli RV• (Named after Swiss scientist Jacob Bernoulli 1654 -1705) Consider a random experiment withsample space S and let A, A ⊂ S be an event. Let X = 1 if A happens and X = 0 if Adoes not happen. Suppose P (A) = p and P (Ac) = q = 1 − p . The parameters p and q arecalled the probability of “success” and “failure”, respectively. The RV X is Bernoulli andthe possible values of X belong to Sx= {0, 1} (Sxis finite).• The pmf of a Bernoulli RV is:P (X = 1) = pX(1) = p P (X = 0) = pX(0) = 1 − p• It is easy to verify thatPk∈SxP (X = k) = p + (1 − p) = 1• The CDF of a Bernoulli RV is:FX(x) = P (X ≤ x) =0 x < 01 − p 0 ≤ x < 11 x ≥ 1• Bernoulli RV is a good model for any random experiment with two possible outcomes, forexample, yes/no answer (of a respondent in an opinion poll), died/survived (in a drug trial),heads/tails (flipping a coin), error/error-free (in communications), defective/non-defective (ina manufacturing quality test), diseased/healty (in a blood test), boy/girl (a newborn baby).EXAMPLE 1We randomly select a chip and test it in a manufactory. Let A be the event that the selectedchip is not defective and P (A) = p. Define the RV X such that X = 1 when the chip is notdefective and X = 0 when the chip is defective. What are Sx, CDF, pmf of X ?EXAMPLE 2Consider a binary symmetric channel (BSC) with binary input X and binary output Y , andtransition error probability of ². Suppose the inputs are equally probable, i.e., P (X = 0) =P (X = 1) = 1/2. What are Sxand Sy? What are CDF and pmf of X? What are CDF andpmf of Y ?F Binomial RV• Suppose a random experiment is repeated n times. Let X be the number of times that anevent A happens in n trials, and in each trial P (A) = p. The RV X is Binomial, denoted asX ∼ B(n, p), and the possible values of X belong to Sx= {0, 1, ..., n} (Sxis finite).• The pmf of a Binomial RV is:P (X = k) = pX(k) =µnk¶pk(1 − p)n−kfor k = 0, 1, ..., n• We havePk∈SxP (X = k) =Pnk=0µnk¶pk(1 − p)n−k= (p + 1 − p)n= 1 due to Binomialtheorem.• The CDF of a Binomial RV is:FX(x) = P (X ≤ x) =0 x < 0pX(0) 0 ≤ x < 1pX(0) +pX(1) 1≤x <2pX(0) + pX(1) + pX(2) 2 ≤ x < 3... ...pX(0) + pX(1) + pX(2) + ... + pX(n − 1) n − 1 ≤ x < n1 x ≥ nor equivalently (in a more compact way):FX(x) = P (X ≤ x) =0 x < 0Pbxci=0pX(i) =Pbxci=0µni¶pi(1 − p)n−i0 ≤ x < n1 x ≥ nwhere bxc is the nearest integer to x such that bxc ≤ x.Page 2• For an integer k, 0 ≤ k ≤ n we have P (X ≤ k) =Pki=0µni¶pi(1 − p)n−i.• Binomial distribution is used to describe an experiment with n trials, for which the proba-bility of success p is the same for each trial and each trial has only two possible outcomes.Examples where Binomial distribution arises: number of “yes” answer in an opinion poll(poll has a single “yes/no” question) collected from n people, number of survivors/deaths ina conducted clinical drug trial involving n people, number of heads when flipping a coin ntimes, number of erroneous received packets in a wireless communication system among ntransmitted packets, number of defective chips in a manufacturing quality test performed onn chips, number of diseased/healty persons based on a blood test involving n people, ... .EXAMPLE 3We randomly select n chips and test them in a manufactory. Let A be the event that a chip isnot defective and P (A) = p. Let X be the number of non-defective chips among the selectedn chips. What are Sx, CDF, pmf of X ?EXAMPLE 4The BSC is used to transmit data packets of length n bits. When sending a packet the channelmay introduce an error for each bit with probability ². Let X be the number of errors in areceived packet. What are Sx, CDF, pmf of X ?EXAMPLE 5A system uses redundancy for reliability: three microprocessors are installed and the systemis designed so that it operates as long as at least one of the microprocessors is functioning.Suppose that the probability that a microprocessor is still functioning after t seconds isp = e−λt, and each of the three components may fail independently. What is the probabilitythat the system is still functioning after t seconds?F Geometric RV• Suppose we count the number M of independent Bernoulli trials until the first occurrence ofa success (i.e., the first time that X = 1). The RV M is Geometric and Sm= {1, 2, 3, ...}(Smis countable, i.e., discrete and infinite).• The pmf of a Geometric RV is:P (M = k) = pM(k) = (1 − p)k−1p k = 1, 2, 3, ...Page 3• Geometric RV is the only discrete RV that satisfies the memoryless property:P (M ≥ k + j|M > j) = P (M ≥ k) for all j , k > 1• We havePk∈SmP (M = k) =P∞k=1qk−1p = p∞Xk=1qk−1| {z }= p11−q= 1, in which we used thefact thatP∞k=1ak−1=P∞j=0aj=11−afor 0 < a < 1.• The CDF of a Geometric RV is:FM(x) = P (M ≤ x) =0 x < 1pM(1) 1 ≤ x < 2pM(1) + pM(2) 2 ≤ x < 3pM(1) + pM(2) + pM(3) 3 ≤ x < 4... ...pM(1) + pM(2) + pM(3) + ... + pM(n − 1) n − 1 ≤ x < n... ...or equivalently (in a more compact way):FX(x) = P (M ≤ x) =(0 x < 1Pbxci=1pM(i) =Pbxci=1qi−1p = 1 − qbxc1 ≤ x < ∞where bxc is the nearest integer to x such that bxc ≤ x.• For an integer k, 1 ≤ k < ∞ we have P (M ≤ k) =Pki=1qi−1p = pPk−1j=0qj= p1−qk1−q= 1−qk,in which we used the fact thatPN−1j=0aj=1−aN1−afor 0 < a < 1 and a positive integer N .• Typical situations where Geometric RV arises are: number of times we need to retransmit apacket until it is received correctly, number of times we need to flip a coin until we observethe first heads, ... .EXAMPLE 6Every time we send a packet over a noisy channel, there is a probability of ε that the packetgets corrupted (i.e., 1 − ε is the probability that the packet is received correctly). Let Y bethe number of times a data packet