Question 2

Let X be a continuous random variablewith 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 sampleu1=.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 probabilitythat exactly 2 out of 20 will have values between 25 and 30 when they fail?

The continuous random variable is generated with a pdf function.