[Joint PDFS will be covered in Week 7] Let the random variables X and Y be the portions of the time in a day that two alternative routes between Topkapi and Uskudar have congestion (X for Route 1 and Y for Route 2). The joint PDF is given by xyx.y) = 2x2 + y² where 0sxys1, (b) Assume Z=X +Y and W= XY and traffic experts are interested in the expected values of Z and W. E(Z) =D and E(W) =D (Simplify your answers. Do not convert fractions into decimals.) (C) Find the variances of X andY as well as the covariance between them. VX) =D. vY) =D and CovX, Y) =D (Simplify your answers. Do not convert fractions into decimals.) (d) The variance of Z can be found as V(Z) =. (Simplify your answer. Do not convert fractions into decimals.)
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![[Joint PDFS will be covered in Week 7] Let the random variables X and Y be the portions of the time in a day that two alternative routes between Topkapi and Uskudar have congestion (X for Route 1 and Y for Route 2). The joint PDF is given by
fxy(x.y) = 2x2 + y? where Osxys1.
(b) Assume Z=X+Y and W= XY and traffic experts are interested in the expected values of Z and W.
E(Z) = and E(W) =O
(Simplify your answers. Do not convert fractions into decimals.)
(c) Find the variances of X and Y as well as the covariance between them.
v(X) =D VY) =D and Cov(X, Y) =D
(Simplify your answers. Do not convert fractions into decimals.)
(d) The variance of Z can be found as V(Z) =-
(Simplify your answer. Do not convert fractions into decimals.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b5cc065-d767-4a47-b400-aded8f835cee%2Fc9e3d28a-9212-4f1a-9af9-1a4ec8c1f794%2Fpxsqvbc_processed.jpeg&w=3840&q=75)
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