Hi
Can anyone help me with this little lot?
1.A certain random variable has a probability density function of the
form
fX(x) = ce−.2xu(x). Find the following:
(a) the constant c,
(b) Pr(X > 2),
(c) Pr(X < 3),
(d) Pr(X < 3|X > 2).
2.Using the normalization integral for a Gaussian random variable,
find an
analytical expression for the following integral:
I = ∞
−∞
exp(−h(ax2 + bx + c)) dx,
where a > 0, b, and c are constants
3. Imagine that you are trapped in a circular room with three doors
symmetrically
placed around the perimeter. You are told by a mysterious voice
that one door leads to the outside after a two-hour trip through a
maze.
However, the other two doors lead to mazes that terminate back in the
room after a two-hour trip, at which time you are unable to tell
through
which door you exited or entered. What is the average time for escape
to the outside? Can you guess the answer ahead of time? If not, can
you
provide a physical explanation for the answer you calculate?
4.Let X be a Gaussian random variable with zero mean and arbitrary
variance, Γ蓋子2. Given the transformation Y = X3, find fY(y).
5.Suppose X is uniformly distributed over (0, 1). Using the results of
the
previous problem, find transformations Y = g(X) to produce random
variables with the following distributions:
(a) exponential,
(b) Rayleigh,
(c) Cauchy,
(d) geometric,
(e) Poisson.
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