Internet Investments in the 1990s The following excerpt is from an article in The New York Times in July 1999: 35 While statistics are not available for web entrepreneurs who fail, the venture capitalists that finance such Internet start-up companies have a rule of thumb. For every 10 ventures that receive financing—and there are plenty that do not—2 will be stock market successes, which means spectacular profits for early investors; 3 will be sold to other concerns, which translates into more modest profits; and the rest will fail. a. What is a sample space for the scenario? b. Write down the associated probability distribution. c. What is the probability that a start-up venture that receives financing will realize profits for early investors?
Internet Investments in the 1990s The following excerpt is from an article in The New York Times in July 1999: 35 While statistics are not available for web entrepreneurs who fail, the venture capitalists that finance such Internet start-up companies have a rule of thumb. For every 10 ventures that receive financing—and there are plenty that do not—2 will be stock market successes, which means spectacular profits for early investors; 3 will be sold to other concerns, which translates into more modest profits; and the rest will fail. a. What is a sample space for the scenario? b. Write down the associated probability distribution. c. What is the probability that a start-up venture that receives financing will realize profits for early investors?
Solution Summary: The author explains that a sample space is the set of all possible outcomes for an experiment.
Internet Investments in the 1990s The following excerpt is from an article in The New York Times in July 1999:35
While statistics are not available for web entrepreneurs who fail, the venture capitalists that finance such Internet start-up companies have a rule of thumb. For every 10 ventures that receive financing—and there are plenty that do not—2 will be stock market successes, which means spectacular profits for early investors;
3 will be sold to other concerns, which translates into more modest profits; and the rest will fail.
a. What is a sample space for the scenario?
b. Write down the associated probability distribution.
c. What is the probability that a start-up venture that receives financing will realize profits for early investors?
Definition Definition For any random event or experiment, the set that is formed with all the possible outcomes is called a sample space. When any random event takes place that has multiple outcomes, the possible outcomes are grouped together in a set. The sample space can be anything, from a set of vectors to real numbers.
(3) (16 points) Consider
z = uv,
u = x+y,
v=x-y.
(a) (4 points) Express z in the form z = fog where g: R² R² and f: R² →
R.
(b) (4 points) Use the chain rule to calculate Vz = (2, 2). Show all intermediate
steps otherwise no credit.
(c) (4 points) Let S be the surface parametrized by
T(x, y) = (x, y, ƒ (g(x, y))
(x, y) = R².
Give a parametric description of the tangent plane to S at the point p = T(x, y).
(d) (4 points) Calculate the second Taylor polynomial Q(x, y) (i.e. the quadratic
approximation) of F = (fog) at a point (a, b). Verify that
Q(x,y) F(a+x,b+y).
=
(6) (8 points) Change the order of integration and evaluate
(z +4ry)drdy .
So S√ ²
0
(10) (16 points) Let R>0. Consider the truncated sphere S given as
x² + y² + (z = √15R)² = R², z ≥0.
where F(x, y, z) = −yi + xj .
(a) (8 points) Consider the vector field
V (x, y, z) = (▼ × F)(x, y, z)
Think of S as a hot-air balloon where the vector field V is the velocity vector
field measuring the hot gasses escaping through the porous surface S. The flux
of V across S gives the volume flow rate of the gasses through S. Calculate
this flux.
Hint: Parametrize the boundary OS. Then use Stokes' Theorem.
(b) (8 points) Calculate the surface area of the balloon. To calculate the surface
area, do the following:
Translate the balloon surface S by the vector (-15)k. The translated
surface, call it S+ is part of the sphere x² + y²+z² = R².
Why do S and S+ have the same area?
⚫ Calculate the area of S+. What is the natural spherical parametrization
of S+?
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