De Anza College Probability Density Function Questions

Question Description

1.a) For what value of the constant LaTeX: c $c$ is the following function a probability density function?

$LaTeX: displaystylef(x) = left{ begin{array}{ll} frac{c}{x^3} & textrm{if }x geq 1\ 0 & textrm{otherwise} end{array} right.$ $f$ ( x ) = { c x 3 if x ? 1 0 otherwise

b) What is the mean, LaTeX: mu $?$ , of this probability density function?

2.If T is a region in the xy-plane whose moment about the y-axis is given by the definite integral

LaTeX: displaystyleM_y = rho int_0^1 left( x - x^2 right) dx $M$ y = ? ? 0 1 ( x ? x 2 ) d x

(where LaTeX: rho $?$ is the density), what might T look like? Sketch a possible region T. Be sure to label your sketch, including the equations of any boundary curves. Do not evaluate the integral.

3.A 40-foot rope with a linear density of 0.1 pounds per foot is hanging from the edge of a balcony. A 20-pound weight is attached to the end of the rope. How much work does it take to lift the rope and the weight up to the level of the balcony?

4.Evaluate the integral:

$LaTeX: displaystyleint_0^1 frac{x^3}{sqrt{4 - x^2}} , dx$

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