WorksheetI
.pdf
School
University of Pennsylvania**We aren't endorsed by this school
Course
MATH 1400
Subject
Mathematics
Date
Jan 14, 2025
Pages
3
Uploaded by LieutenantSwanPerson1246
Worksheet IThe learning objectives for this unit are:Recall and apply the definition of a centroidIdentify the centroid with the center of balance (assuming constant density)Informally justify the formulas for the coordinates of the centroid of a regionDetermine the appropriate variable of integration to find the centroid or center of massfor a given regionSet up and evaluate integrals to compute centroidsUse symmetry, when appropriate, to determine one or more coordinates of a centroid orcenter of massExplain the relationship between centers of mass and (weighted-) averagesSet up and evaluate integrals to compute centers of mass of two- and three-dimensionalfigures with constant and non-constant densityIdentify an appropriate measure (e.g. number, length, area, volume) for computing prob-abilities in contextCompute geometric and discrete probabilities from contextRecall the definition of a probability density functionExplain the relationship between density functions and probabilities using integrationIdentify probability density functions, and modify functions to create probability densityfunctionsReason about pdfs from probabilities, and vice versaUse techniques of integration and/or improper integrals to compute probabilitiesSet up and evaluate an integral to model a scenario concerning probabilityIdentify when a uniform, exponential, or normal distribution might be an appropriatemodel for a random phenomenonRecall and apply the uniform distribution to model random phenomenaApply exponential and normal distributions to model random phenomenaRelate normal distributions to the standard normal distribution via substitution(1) Find the centroid of the region between the parabolay=x2and the liney= 4.1
(2) Consider the region betweeny=mxand thex-axis on the interval [0, c] for some constantsm, c >0.(a) Find the average value of the functiony=mx.(b) Can you interpret the result from (a) geometrically?(c) Find the centroid of the region.(3) Find the value ofCifρ(x) =(6x3x≥C0x < Cis a probability density function.(4) A frustrated Math 1400 student throws a dart at the graph of the region bounded abovebyy= 1 + sin(x) and below byy=xπ, fromx= 0 tox=π.The student’s aim isunpredictable, so that the dart is as likely to land in any portion of the region as anyother portion with the same area. (Despite the imprecise aim, you may assume that thestudent is guaranteed to hit the region, perhaps by repeatedly throwing the dart until itlands there.)(a) Sketch the described region. What is the range of possiblexcoordinates? What isthe range of possibleycoordinates?(b) Find the expectedx-value of the dart’s landing spot.(c) Find the probability that thex-coordinate of a dart is greater than or equal to 2.(d) Find the probability that they-coordinate of a dart is less than or equal to 1.(5) A certain random variable measures the time in months until Prof G’s laptop crashes. Itsdensity function isρ(x) =(16e−x/6x≥00x <0Find the probability that the laptop does not crash in the first year.(6) Letρ(x) =1π11 +x2be a probability density function from the random variableXon(−∞,∞).(a) Find the probability that the random variable is less than or equal to 1.(b) Find the probability that the random variable is between 0 and 1.(c) Can you explain geometrically why (b) and (c) are asking essentially the same thing?2
(7) The function,ρ(x), graphed below is a piecewise linear function on [0,1].xis a vari-able representing a number generated by a computer between the value of 0 and 1 andρ(x) is the probability density functionfor generated numbers.(a) What doesP([13,12]) represent?(b) CalculateP([13,12]).(8) For what value ofkis the functionf(n) =k6n, wherenranges over the positive integers,a probability distribution?(9) Letρ(r) =(Cr2e−2r/br≥00r <0FindCso that it is a probability density function (pdf) for the random variabler. (Theletterbis a constant). This is used to model the distance between the nucleus and theelectron in a hydrogen atom. Withb >0, it is called the Bohr length.3