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Take Home Test 3

Score: 50

1.   A probability density function meets two requirements:

1) The area under the curve is 1; 2) the curve is always positive.

a.    The Weibull distribution has the probability density function 0 (for negative x) and

                          (for nonnegative x).

The parameters k and λ are positive real numbers, defining the shape and scale of the curve respectively.

Show that this curve meets the requirements for a probability density function. (6)

b.    For k = 2, λ = 4, find the probability (i.e. the area) that x lies between 2 and 4. (4)

c.    For k = 1, this distribution is also called the exponential distribution.

i.   Find                                                  (also known as the mean) of the exponential

distribution. (5)

ii.   Find                                                                        (also known as the standard

deviation) of the exponential distribution. (5)

2.    For the curve                                                                     :

a.    Find the length of the curve over the stated x-interval. (5)

b.    Find the surface area of the solid of revolution obtained by revolving the region represented by the integral about the y-axis. (5)

3.    Solve the following:

a.                                         (5)

b.                                              (5)

c.    The area between the parabola y = ½ x2  and the circle x2+y2  = 8, where the circle is above the parabola (5)

d.                                                 (5)