Solve the following:
Q1: Let X be a continuous random variable with the pdf f(x) = 42x5(1-x) 0
Q2: 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).
Q3: 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?