tut7ans - are independent,0<x 1<1 zero...

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COMP211 Data and System Modeling (PROB/STAT) Tutorial (Week 7) Answer 1) For continuous random variables X and Y with the joint probability density function, a) Find ,,, b) Find the covariance of X, Y cov(X,Y) and the correlation coefficient of X and Y. Answer: a) =11/144 =11/144 b) =-1/144 =-1/11 2) Let random variables X1 and X2 have the joint pdf, a) Find marginal pdfs of X 1 and X 2 b) Find the conditional pdf of X 2 given X 1 = x 1 c) Find the conditional pdf of X 1 given X 2 = x 2 d) Find the conditional expectation and variance of X 2 given X 1 = x 1 e) Find the conditional expectation and variance of X 1 given X 2 = x 2 f) Find P(0< X 1 <1/2) and P(0< X 1 <1/2| X 2 =3/4) Answer: a) , 0<x 1 <1; zero elsewhere , 0<x 2 <1; zero elsewhere b) ,x 1 < x 2 <1; zero elsewhere c) ,0<x 1 < x 2 ; zero elsewhere d) ,0<x 1 <1 ,0<x 1 <1 e) ,0<x 2 <1 ,0<x 2 <1 f)
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3) Show that the random variables X1 and X2 with joint pdf,
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Unformatted text preview: are independent. ,0<x 1 <1; zero elsewhere ,0<x 2 <1; zero elsewhere Since , so the random variables X1 and X2 are independent. 4) One cookie in 5 is broken. If 20 are grabbed, what are (a) P(4 broken), (b) P(at most 2 broken), and (c) P(at least 4 broken)? Answer: The situation is analogous to a succession of Bernoulli trails, with a broken cookie corresponding to a success. Let X be the number of broken cookies. Hence p=1/5, q=1-p=4/5, and n=20. Consequently, a) P(4 broken) = P(X=4) = = 0.2182 b) P(at most 2 broken) = P(X= 0 or 1 or 2) = = 0.2061 c) P(at least 4 broken) = 1 – P(X<4) = 1 – ( P(X=0) +P(X=1) +P(X=2) +P(X=3) ) = = 0.5886...
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tut7ans - are independent,0<x 1<1 zero...

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