Continuous random variable generated

Question 2

Let X be a continuous random variable with the pdf f(x) =
42×5(1-x) 0<x<1 and let Y=X3. Suppose u=.145 is a random number generated
from the uniform (0,1). Determine the corresponding random number from the
distribution Y. This will involve solving a nonlinear equation by using
Newton’s Method.

Question 3

Let F be the cdf of an integer-valued random variable X and
U be the uniform random variable on the interval (0,1). Let Y=k if F(k-1) <
U < F(k).

a. Show that
the cdf of Y is F.

b. Use part
(a) to show how to generate the geometric random variable (number of Bernoulli
trials until to get first success) with p=.5 using uniform (0,1) random
variable. Compute the corresponding values of X using p=.5 and a random sample
u1=.981, u2=.671, u3=.078 from U(0,1).

Question 4

Let T be the lifetime of a device and assume that the pdf of
T is given by f(t)=(1/9)te-t/3 for t>0. The value of the device is 9 if it
fails before time t=2, otherwise it has value V=5t.

a. Find the
cdf of V

b. If we purchase
20 such devices, what is the probability that exactly 2 out of 20 will have
values between 25 and 30 when they fail?

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