Fundamentals of Probability and Statistics for Engineers

156 Fundamentals of Probability and Statistics for Engineers5.10 At a given location, the PDF of annual wind speed, V, in miles per hour is foundto beThe wind force W exerted on structures is proportional to V 2 .Let W ˆ aV 2 .(a) Determine the pdf of W and its mean and variance by using f W (w).(b) Determine the mean and variance of W directly from the knowledge of F V (v).5.11 An electrical device called a full-wave rectifier transforms input X to the device, tooutput Y according to Y ˆjX j. If input X has a pdf of the form" #f X …x† ˆ 1…2† exp …x 1† 2; 1 < x < 1;1=2 2(a) Determine the pdf of Y and its mean and variance using f Y (y).(b) Determine the mean and variance of Y directly from the knowledge of f X (x).5.12 An electrical device gives output Y in terms of input X according to 1; for X > 0;Y ˆ g…X† ˆ0; for X 0:Is random variable Y continuous or discrete?Determine its probability distributionin terms of the pdf of X.5.13 The kinetic energy of a particle with mass m and velocity V is given bySuppose that m is deterministic and V is random with pdf given byf V …v† ˆ av2 e bv2 ; for v > 0;0; elsewhere:Determine the pdf of X.8 F V …v† ˆ exp v 0;36:6:0; elsewhere:X ˆ mV225.14 The radius R of a sphere is known to be distributed uniformly in the range0: 99 r0 r 1:01r 0. Determine the pdfs of (a) its surface area and (b) of itsvolume.5.15 A resistor to be used as a component in a simple electrical circuit is randomlychosen from a stock for which resistance R has the pdff R …r† ˆ a2 re ar ; for r > 0;0; elsewhere:TLFeBOOK