De Anza College Probability Density Function Questions

1. **a)** For what value of the constant ![LaTeX: c]( is the following function a probability density function? ![LaTeX: \displaystyle f(x) = \left\{ \begin{array}{ll} \frac{c}{x^3} & \textrm{if }x \geq 1\\ 0 & \textrm{otherwise} \end{array} \right.]( ( x ) = { c x 3 if x ≥ 1 0 otherwise **b)** ****What is the mean,![LaTeX: \mu]( μ, 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: \displaystyle M_y = \rho \int_0^1 \left( x – x^2 \right) dx]( y = ρ ∫ 0 1 ( x − x 2 ) d x (where ![LaTeX: \rho](ρ 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: \displaystyle \int_0^1 \frac{x^3}{\sqrt{4 – x^2}} \, dx](

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