ROCHESTER ECE 270 - Discrete RVs Study Notes (6 pages)

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Discrete RVs Study Notes



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Discrete RVs Study Notes

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Pages:
6
School:
University of Rochester
Course:
Ece 270 - Introduction to Probability
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ECE270 Handout 6 Discrete 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 behavior Important discrete RVs are Bernoulli RV Binomial RV Geometric RV Poisson RV and discrete uniform RV F Bernoulli RV Named after Swiss scientist Jacob Bernoulli 1654 1705 Consider a random experiment with sample space S and let A A S be an event Let X 1 if A happens and X 0 if A does not happen Suppose P A p and P Ac q 1 p The parameters p and q are called the probability of success and failure respectively The RV X is Bernoulli and the possible values of X belong to Sx 0 1 Sx is finite The pmf of a Bernoulli RV is P X 1 pX 1 p It is easy to verify that P k Sx P X 0 pX 0 1 p P X k p 1 p 1 The CDF of a Bernoulli RV is x 0 0 1 p 0 x 1 FX x P X x 1 x 1 Bernoulli RV is a good model for any random experiment with two possible outcomes for example 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 in a manufacturing quality test diseased healty in a blood test boy girl a newborn baby EXAMPLE 1 We randomly select a chip and test it in a manufactory Let A be the event that the selected chip is not defective and P A p Define the RV X such that X 1 when the chip is not defective and X 0 when the chip is defective What are Sx CDF pmf of X EXAMPLE 2 Consider a binary symmetric channel BSC with binary input X and binary output Y and transition error probability of Suppose the inputs are equally probable i e P X 0 P X 1 1 2 What are Sx and Sy What are CDF and pmf of X What are CDF and pmf of Y F Binomial RV Suppose a random experiment is repeated n times Let X be the number of times that an event A happens in n trials and in each trial P A p The RV X is Binomial denoted as X B n p and the possible values of X belong to Sx 0 1 n Sx is finite The pmf of a Binomial RV is n k P X k pX k We have P k Sx P X k Pn k 0 n k pk 1 p n k for k 0 1 n pk 1 p n k p 1 p n 1 due to Binomial theorem The CDF of a Binomial RV is 0 p X 0 pX 0 pX 1 pX 0 pX 1 pX 2 FX x P X x pX 0 pX 1 pX 2 pX n 1 1 x 0 0 x 1 1 x 2 2 x 3 n 1 x n x n or equivalently in a more compact way 0 x 0 P P n bxc bxc pi 1 p n i 0 x n FX x P X x i 0 pX i i 0 i 1 x n where 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 Pk i 0 n i pi 1 p n i Binomial distribution is used to describe an experiment with n trials for which the probability 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 in a conducted clinical drug trial involving n people number of heads when flipping a coin n times number of erroneous received packets in a wireless communication system among n transmitted packets number of defective chips in a manufacturing quality test performed on n chips number of diseased healty persons based on a blood test involving n people EXAMPLE 3 We randomly select n chips and test them in a manufactory Let A be the event that a chip is not defective and P A p Let X be the number of non defective chips among the selected n chips What are Sx CDF pmf of X EXAMPLE 4 The BSC is used to transmit data packets of length n bits When sending a packet the channel may introduce an error for each bit with probability Let X be the number of errors in a received packet What are Sx CDF pmf of X EXAMPLE 5 A system uses redundancy for reliability three microprocessors are installed and the system is 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 is p e t and each of the three components may fail independently What is the probability that 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 of a success i e the first time that X 1 The RV M is Geometric and Sm 1 2 3 Sm is countable i e discrete and infinite The pmf of a Geometric RV is P M k pM k 1 p k 1 p Page 3 k 1 2 3 Geometric RV is the only discrete RV that satisfies the memoryless property P M k j M j P M k We have fact that P k Sm P M k P k 1 k 1 a P j 0 a The CDF of a Geometric RV FM x P M x j P k 1 p p k 1 q 1 1 a X for all j k 1 1 1 in which we used the q k 1 p 1 q k 1 z for 0 a 1 is 0 pM 1 pM 1 pM 2 pM 1 pM 2 pM 3 pM 1 pM 2 pM 3 pM n 1 x 1 1 x 2 2 x 3 3 x 4 n 1 x n or equivalently in a more compact way 0 x 1 Pbxc Pbxc i 1 FX x P M x bxc p 1 q 1 x i 1 pM i i 1 q where bxc is the nearest integer to x such that bxc x P P 1 q k j k For an integer k 1 k we have P M k ki 1 q i 1 p p k 1 j 0 q p …


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