De Anza College Probability Density Function Questions

1. **a)** For what value of the constant ![LaTeX: c](https://deanza.instructure.com/equation_images/c)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.](https://deanza.instructure.com/equation_images/%255Cdisplaystyle%250Af(x)%2520%253D%2520%255Cleft%255C%257B%2520%255Cbegin%257Barray%257D%257Bll%257D%250A%2520%2520%255Cfrac%257Bc%257D%257Bx%255E3%257D%2520%2526%2520%255Ctextrm%257Bif%2520%257Dx%2520%255Cgeq%25201%255C%255C%250A%2520%25200%2520%2526%2520%255Ctextrm%257Botherwise%257D%250A%2520%255Cend%257Barray%257D%2520%255Cright.)f ( x ) = { c x 3 if x ≥ 1 0 otherwise **b)** ****What is the mean,![LaTeX: \mu](https://deanza.instructure.com/equation_images/%255Cmu) 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](https://deanza.instructure.com/equation_images/%255Cdisplaystyle%250AM_y%2520%253D%2520%255Crho%2520%255Cint_0%255E1%2520%255Cleft(%2520x%2520-%2520x%255E2%2520%255Cright)%2520dx)M y = ρ ∫ 0 1 ( x − x 2 ) d x (where ![LaTeX: \rho](https://deanza.instructure.com/equation_images/%255Crho)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](https://deanza.instructure.com/equation_images/%255Cdisplaystyle%250A%255Cint_0%255E1%2520%255Cfrac%257Bx%255E3%257D%257B%255Csqrt%257B4%2520-%2520x%255E2%257D%257D%2520%255C%252C%2520dx)

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