180amid2revsolns - 180 Midterm 2 Review Solutions 1 At the...

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180 Midterm 2 Review Solutions 1) At the mayoral election in a very large city, 51% of the people support candidate A , while 49% support B . Use the normal approximation to the binomial distribution to find a numerical expression for the probability that among 500 people surveyed in this city, 49% are for candidate A while 51% are for candidate B . Answer Let X denote the number of people (surveyed) who are for candidate A . Then X Bin (500 , . 51), E ( X ) = 500( . 51) = 255 and SD ( X ) = 500 × 0 . 51 × 0 . 49 11 . 18. We note that P ( X = 245) = P (245 X 245) Φ parenleftbigg 245 + 1 / 2 - 255 11 . 18 parenrightbigg - Φ parenleftbigg 245 - 1 / 2 - 255 11 . 18 parenrightbigg Φ( - 0 . 84987) - Φ( - 0 . 9393) 0 . 0241 2) In some geographical area, 65% of the trees are oak trees. Suppose that in your field work you survey 1,000 trees. What are the chances that you identify 700 oak trees? Use the normal approximation to the binomial distribution to find a numerical expression. Answer Let X denote the number of oak trees we find. Then X Bin (1000 , 0 . 65), and SD ( X ) = 1000 × 0 . 65 × 0 . 35 15 . 083. Then P ( X 700) = P (700 X 1000) Φ parenleftbigg 1000 + 1 / 2 - 650 15 . 083 parenrightbigg - Φ parenleftbigg 700 - 1 / 2 - 650 15 . 083 parenrightbigg 1 - Φ(3 . 31496) 0 3) Let f ( x ) = Ce 3 | x | defined for all x ( -∞ , ) Assume f is a probability density function, and let X be a random variable with PDF f . (a) Find C (b) Compute P ( - 2 X 3) (c) Compute E [ X ] and Var( X ). Answer For ( a ) note that integraldisplay −∞ Ce 3 | x | = 2 integraldisplay 0 Ce 3 x dx = - 2 3 Ce 3 x vextendsingle vextendsingle vextendsingle vextendsingle 0 = 2 3 C so C = 3 / 2.
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