exercises and quizzes and paper, Quizzes of Computer Science

Download exercises and quizzes and paper and more Quizzes Computer Science in PDF only on Docsity! Important Families of Discrete Random Variables There are SIX important families of Discrete Random Variables. There is one formula for the PMF of all the random variables in a family. Depending on the family, the PMF formula contains one or two parameters. Bernoulli (p) Random Variable Geometric (p) Random Variable Binomial (n, p) Random Variable Pascal (k, p) Random Variable Discrete Uniform (k, |) Random Variable A w& KR Ww NS Poisson (a) Random Variable 6/1/2021 Capital University of Science & Technology, Islamabad; Probability & Random Variables (EE2413) 1 Bernoulli (p) Discrete Random Variable Definition 2.5 Bernoulli (p) Random Variable X is a Bernoulli (p) random variable if the PMF of X has the form l-—p x=0 ~ Py(xy)=} por x=l1w 0 otherwise where the parameter p is in the range 0 < p < 1. Suppose you test one circuit. With probability p, the circuit is rejected. Let X be the number of rejected circuits in one test. What 1s P(x)? If there is a 0.2 probability of a reject, l—p x=0 1 J Py(x)=4 p x=! 2 os 08 x=0 0 otherwise © v Py()=} 02 x=1 o I 0 otherwise -1 0 1 2 x 6/1/2021 Capital University of Science & Technology, Islamabad; Probability & Random Variables (EE2413) 2 Binomial (n, p) Discrete Random Variable Definition 2.7 Binomial (n, p) Random Variable X is a binomial (n, p) random variable if the PMF of X has the form Py (x)= (“)era — p)"~* where 0 < p < | and n is an integer such thatn > 1. p independent.of the results of other tests. Le qual the number of rejects in thd m)tests. Find the PMF Px(k). If there is a 0.2 probability of a reject and we perform 10 tests, Pek) = (i) Xa _pyrmk % 0.2 Suppose we test 7 circuits and each circuit is is ected with probability (k) P. 0 Px (k= (; Jorkeos!e-t 0 0 5 10 k 6/1/2021 Capital University of Science & Technology, Islamabad; Probability & Random Variables (EE2413) 5 Pascal (k, p) Discrete Random Variable oo 2.8 Pascal (k, p) Random Variable Xisa. is a Pascal (k, p) random variable if the PMF of X has the form al x-l\ ¥% [C0 POWweny Ae\ Py (x) = (yr a — Pp L ») e-D p a? where 0 < p < | andk is an integer such thatk > 1. 2p Lf Ny For a sequence of 7 independent trials with success probability p,a Pascal random variable is the number of trials up to and including the &th suécess) We must keep in mind that for a Pascal “ “P) random variable X , Px(x) is nonzero only forx =k, k+1,. If there is a 0.2 probability of a reject and we seek four defective circuits, the random variable L is the number of tests necessary to find the four circuits. The PMF is “ (ss on i-1 = 0.02 P= ( 3 Jo2r*osy 4° (2.23) PW 0 10 20 30 40 1 6/1/2021 Capital University of Science & Technology, Islamabad; Probability & Random Variables (EE2413) 6 6/1/2021 Capital University of Science & Technology, Islamabad; Probability & Random Variables (EE2413) 7 [ths ., 6,4,9,% \0 | F a “ x rat vost wowed) = PUI Ae CL) (ried be ed eB ig 2th 6/1/2021 Capital University of Science & Technology, Islamabad; Probability & Random Variables (EE2413) 10 Related Problems - Poisson (a) Random Variable Definition 2.10 Poisson (a) Random Variable X is a Poisson (a) random variable if the PMF of X has the form a*e~*/x! x =0,1,2,..., 0 otherwise, Py(x)= where the parameter a is in the range a > 0. Example: The number of hits at a Web site in any time interval is a Poisson random variable. A particular site has on average / = 2 hits per second. What 1s the probability that there are no hits in an interval of 0.25 seconds? What is the probability that there are no more than two hits in an interval of one second? 0.54e-°5 ht! h =0,1,2,... Puy =) 9 otherwise. 6/1/2021 Capital University of Science & Technology, Islamabad; Probability & Random Variables (EE2413) 11 Related Problems - Poisson (a) Random Variable Definition 2.10 Poisson (a) Random Variable X is a Poisson (a) random variable if the PMF of X has the form a*e~*/x! x =0,1,2,..., Px (x) = 0 otherwise, where the parameter a is in the range a > 0. Example: Calls arrive at random times at a telephone switching office with an average of 2 = 0.25 calls per second. The PMF of the number of calls that arrive in any 7= 20-second interval is the Poisson (5) random variable. 0.2 0.1 P;(j) = SJe~S/j! fj =0,1,..., 0 otherwise. 0 0 5 10 15 Jj 6/1/2021 Capital University of Science & Technology, Islamabad; Probability & Random Variables (EE2413) 12