Math 302, Assignment 6 Due March 9
1. Let X be a random variable with p.d.f.
f(x) =2x 2 x> 2
0 otherwise
(a) Compute the c.d.f. of X.
(b) Find P(X > 3).
(c) Find P(X > 3jX 3).
(d) P(X 1).
(e) P(X 1):
5. Let X N( 1;1):
(a) Find c such that P(X >c) = 13:
(b) Compute EX2.
6. Suppose a random number generator generates 20 numbers per second, where
each number is drawn uniformly from the interval [9,10], independently of all other
numbers. We are interested in the event that one of the drawn numbers is very
close to Usain Bolt’s 100m sprint world record, that is, that this number belongs
to the interval [9:575;9:585].
(a) Suppose that the random number generator runs for 10 seconds. Use the Poisson
approximation to estimate the probability that it produces more than 4 numbers
that fall in that interval.
(b) We now let the generator run inde nitely. Use the exponential random variable
to estimate the probability that it produces the rst such number before Usain Bolt
nishes his run, i.e. within 9.58 seconds.
(c) Suppose we let the random number generator compete against Usain Bolt,
as described in (b), every day for 100 days. Use the normal random variable to
approximate the probability that Usain Bolt wins at least 12 of these competitions.
7. Let X N( ; 2), and, for a;b2R, de ne the random variable Y = aX + b.
Show that Y N(a +b;a2 2). Hint: Which overall strategy did you use in Q2?
8*. Let X be a standard normal random variable. Compute EXn for all n2N